diff --git a/numpy/add_newdocs.py b/numpy/add_newdocs.py
index 09f4e40c4d7a..3916d13045e7 100644
--- a/numpy/add_newdocs.py
+++ b/numpy/add_newdocs.py
@@ -6728,6 +6728,51 @@ def luf(lamdaexpr, *args, **kwargs):
     53
     """)
 
+add_newdoc('numpy.core.multiarray', 'normalize_axis_index',
+    """
+    normalize_axis_index(axis, ndim)
+
+    Normalizes an axis index, `axis`, such that is a valid positive index into
+    the shape of array with `ndim` dimensions. Raises an AxisError with an
+    appropriate message if this is not possible.
+
+    Used internally by all axis-checking logic.
+
+    .. versionadded:: 1.13.0
+
+    Parameters
+    ----------
+    axis : int
+        The un-normalized index of the axis. Can be negative
+    ndim : int
+        The number of dimensions of the array that `axis` should be normalized
+        against
+
+    Returns
+    -------
+    normalized_axis : int
+        The normalized axis index, such that `0 <= normalized_axis < ndim`
+
+    Raises
+    ------
+    AxisError
+        If the axis index is invalid, when `-ndim <= axis < ndim` is false.
+
+    Examples
+    --------
+    >>> normalize_axis_index(0, ndim=3)
+    0
+    >>> normalize_axis_index(1, ndim=3)
+    1
+    >>> normalize_axis_index(-1, ndim=3)
+    2
+
+    >>> normalize_axis_index(3, ndim=3)
+    Traceback (most recent call last):
+    ...
+    AxisError: axis 3 is out of bounds for array of dimension 3
+    """)
+
 ##############################################################################
 #
 # nd_grid instances
diff --git a/numpy/core/_internal.py b/numpy/core/_internal.py
index 741c8bb5fbcb..d73cdcc55ab6 100644
--- a/numpy/core/_internal.py
+++ b/numpy/core/_internal.py
@@ -630,3 +630,6 @@ def _gcd(a, b):
 # Exception used in shares_memory()
 class TooHardError(RuntimeError):
     pass
+
+class AxisError(ValueError, IndexError):
+    pass
diff --git a/numpy/core/numeric.py b/numpy/core/numeric.py
index 97d19f008a78..066697f3eda6 100644
--- a/numpy/core/numeric.py
+++ b/numpy/core/numeric.py
@@ -17,7 +17,7 @@
     inner, int_asbuffer, lexsort, matmul, may_share_memory,
     min_scalar_type, ndarray, nditer, nested_iters, promote_types,
     putmask, result_type, set_numeric_ops, shares_memory, vdot, where,
-    zeros)
+    zeros, normalize_axis_index)
 if sys.version_info[0] < 3:
     from .multiarray import newbuffer, getbuffer
 
@@ -27,7 +27,7 @@
                     ERR_DEFAULT, PINF, NAN)
 from . import numerictypes
 from .numerictypes import longlong, intc, int_, float_, complex_, bool_
-from ._internal import TooHardError
+from ._internal import TooHardError, AxisError
 
 bitwise_not = invert
 ufunc = type(sin)
@@ -65,7 +65,7 @@
     'True_', 'bitwise_not', 'CLIP', 'RAISE', 'WRAP', 'MAXDIMS', 'BUFSIZE',
     'ALLOW_THREADS', 'ComplexWarning', 'full', 'full_like', 'matmul',
     'shares_memory', 'may_share_memory', 'MAY_SHARE_BOUNDS', 'MAY_SHARE_EXACT',
-    'TooHardError',
+    'TooHardError', 'AxisError'
     ]
 
 
@@ -1527,15 +1527,12 @@ def rollaxis(a, axis, start=0):
 
     """
     n = a.ndim
-    if axis < 0:
-        axis += n
+    axis = normalize_axis_index(axis, n)
     if start < 0:
         start += n
     msg = "'%s' arg requires %d <= %s < %d, but %d was passed in"
-    if not (0 <= axis < n):
-        raise ValueError(msg % ('axis', -n, 'axis', n, axis))
     if not (0 <= start < n + 1):
-        raise ValueError(msg % ('start', -n, 'start', n + 1, start))
+        raise AxisError(msg % ('start', -n, 'start', n + 1, start))
     if axis < start:
         # it's been removed
         start -= 1
@@ -1554,7 +1551,7 @@ def _validate_axis(axis, ndim, argname):
         axis = list(axis)
     axis = [a + ndim if a < 0 else a for a in axis]
     if not builtins.all(0 <= a < ndim for a in axis):
-        raise ValueError('invalid axis for this array in `%s` argument' %
+        raise AxisError('invalid axis for this array in `%s` argument' %
                          argname)
     if len(set(axis)) != len(axis):
         raise ValueError('repeated axis in `%s` argument' % argname)
diff --git a/numpy/core/shape_base.py b/numpy/core/shape_base.py
index 70afdb7465ec..58b0dcaac827 100644
--- a/numpy/core/shape_base.py
+++ b/numpy/core/shape_base.py
@@ -5,6 +5,7 @@
 
 from . import numeric as _nx
 from .numeric import asanyarray, newaxis
+from .multiarray import normalize_axis_index
 
 def atleast_1d(*arys):
     """
@@ -347,11 +348,7 @@ def stack(arrays, axis=0):
         raise ValueError('all input arrays must have the same shape')
 
     result_ndim = arrays[0].ndim + 1
-    if not -result_ndim <= axis < result_ndim:
-        msg = 'axis {0} out of bounds [-{1}, {1})'.format(axis, result_ndim)
-        raise IndexError(msg)
-    if axis < 0:
-        axis += result_ndim
+    axis = normalize_axis_index(axis, result_ndim)
 
     sl = (slice(None),) * axis + (_nx.newaxis,)
     expanded_arrays = [arr[sl] for arr in arrays]
diff --git a/numpy/core/src/multiarray/common.h b/numpy/core/src/multiarray/common.h
index 5e14b80a71ca..625ca9d76887 100644
--- a/numpy/core/src/multiarray/common.h
+++ b/numpy/core/src/multiarray/common.h
@@ -134,6 +134,42 @@ check_and_adjust_index(npy_intp *index, npy_intp max_item, int axis,
     return 0;
 }
 
+/*
+ * Returns -1 and sets an exception if *axis is an invalid axis for
+ * an array of dimension ndim, otherwise adjusts it in place to be
+ * 0 <= *axis < ndim, and returns 0.
+ */
+static NPY_INLINE int
+check_and_adjust_axis(int *axis, int ndim)
+{
+    /* Check that index is valid, taking into account negative indices */
+    if (NPY_UNLIKELY((*axis < -ndim) || (*axis >= ndim))) {
+        /*
+         * Load the exception type, if we don't already have it. Unfortunately
+         * we don't have access to npy_cache_import here
+         */
+        static PyObject *AxisError_cls = NULL;
+        if (AxisError_cls == NULL) {
+            PyObject *mod = PyImport_ImportModule("numpy.core._internal");
+
+            if (mod != NULL) {
+                AxisError_cls = PyObject_GetAttrString(mod, "AxisError");
+                Py_DECREF(mod);
+            }
+        }
+
+        PyErr_Format(AxisError_cls,
+                     "axis %d is out of bounds for array of dimension %d",
+                     *axis, ndim);
+        return -1;
+    }
+    /* adjust negative indices */
+    if (*axis < 0) {
+        *axis += ndim;
+    }
+    return 0;
+}
+
 
 /*
  * return true if pointer is aligned to 'alignment'
diff --git a/numpy/core/src/multiarray/conversion_utils.c b/numpy/core/src/multiarray/conversion_utils.c
index c016bb8d10d4..8ed08a366177 100644
--- a/numpy/core/src/multiarray/conversion_utils.c
+++ b/numpy/core/src/multiarray/conversion_utils.c
@@ -259,17 +259,10 @@ PyArray_ConvertMultiAxis(PyObject *axis_in, int ndim, npy_bool *out_axis_flags)
             PyObject *tmp = PyTuple_GET_ITEM(axis_in, i);
             int axis = PyArray_PyIntAsInt_ErrMsg(tmp,
                           "integers are required for the axis tuple elements");
-            int axis_orig = axis;
             if (error_converting(axis)) {
                 return NPY_FAIL;
             }
-            if (axis < 0) {
-                axis += ndim;
-            }
-            if (axis < 0 || axis >= ndim) {
-                PyErr_Format(PyExc_ValueError,
-                        "'axis' entry %d is out of bounds [-%d, %d)",
-                        axis_orig, ndim, ndim);
+            if (check_and_adjust_axis(&axis, ndim) < 0) {
                 return NPY_FAIL;
             }
             if (out_axis_flags[axis]) {
@@ -284,20 +277,16 @@ PyArray_ConvertMultiAxis(PyObject *axis_in, int ndim, npy_bool *out_axis_flags)
     }
     /* Try to interpret axis as an integer */
     else {
-        int axis, axis_orig;
+        int axis;
 
         memset(out_axis_flags, 0, ndim);
 
         axis = PyArray_PyIntAsInt_ErrMsg(axis_in,
                                    "an integer is required for the axis");
-        axis_orig = axis;
 
         if (error_converting(axis)) {
             return NPY_FAIL;
         }
-        if (axis < 0) {
-            axis += ndim;
-        }
         /*
          * Special case letting axis={-1,0} slip through for scalars,
          * for backwards compatibility reasons.
@@ -306,10 +295,7 @@ PyArray_ConvertMultiAxis(PyObject *axis_in, int ndim, npy_bool *out_axis_flags)
             return NPY_SUCCEED;
         }
 
-        if (axis < 0 || axis >= ndim) {
-            PyErr_Format(PyExc_ValueError,
-                    "'axis' entry %d is out of bounds [-%d, %d)",
-                    axis_orig, ndim, ndim);
+        if (check_and_adjust_axis(&axis, ndim) < 0) {
             return NPY_FAIL;
         }
 
diff --git a/numpy/core/src/multiarray/ctors.c b/numpy/core/src/multiarray/ctors.c
index 349b59c5f87d..ee6b66eef461 100644
--- a/numpy/core/src/multiarray/ctors.c
+++ b/numpy/core/src/multiarray/ctors.c
@@ -2793,7 +2793,6 @@ PyArray_CheckAxis(PyArrayObject *arr, int *axis, int flags)
 {
     PyObject *temp1, *temp2;
     int n = PyArray_NDIM(arr);
-    int axis_orig = *axis;
 
     if (*axis == NPY_MAXDIMS || n == 0) {
         if (n != 1) {
@@ -2831,12 +2830,7 @@ PyArray_CheckAxis(PyArrayObject *arr, int *axis, int flags)
         temp2 = (PyObject *)temp1;
     }
     n = PyArray_NDIM((PyArrayObject *)temp2);
-    if (*axis < 0) {
-        *axis += n;
-    }
-    if ((*axis < 0) || (*axis >= n)) {
-        PyErr_Format(PyExc_ValueError,
-                     "axis(=%d) out of bounds", axis_orig);
+    if (check_and_adjust_axis(axis, n) < 0) {
         Py_DECREF(temp2);
         return NULL;
     }
diff --git a/numpy/core/src/multiarray/item_selection.c b/numpy/core/src/multiarray/item_selection.c
index 08b9c5965240..3c0f0782e586 100644
--- a/numpy/core/src/multiarray/item_selection.c
+++ b/numpy/core/src/multiarray/item_selection.c
@@ -1101,16 +1101,12 @@ NPY_NO_EXPORT int
 PyArray_Sort(PyArrayObject *op, int axis, NPY_SORTKIND which)
 {
     PyArray_SortFunc *sort;
-    int axis_orig = axis;
-    int  n = PyArray_NDIM(op);
+    int n = PyArray_NDIM(op);
 
-    if (axis < 0) {
-        axis += n;
-    }
-    if (axis < 0 || axis >= n) {
-        PyErr_Format(PyExc_ValueError, "axis(=%d) out of bounds", axis_orig);
+    if (check_and_adjust_axis(&axis, n) < 0) {
         return -1;
     }
+
     if (PyArray_FailUnlessWriteable(op, "sort array") < 0) {
         return -1;
     }
@@ -1212,17 +1208,13 @@ PyArray_Partition(PyArrayObject *op, PyArrayObject * ktharray, int axis,
     PyArrayObject *kthrvl;
     PyArray_PartitionFunc *part;
     PyArray_SortFunc *sort;
-    int axis_orig = axis;
     int n = PyArray_NDIM(op);
     int ret;
 
-    if (axis < 0) {
-        axis += n;
-    }
-    if (axis < 0 || axis >= n) {
-        PyErr_Format(PyExc_ValueError, "axis(=%d) out of bounds", axis_orig);
+    if (check_and_adjust_axis(&axis, n) < 0) {
         return -1;
     }
+
     if (PyArray_FailUnlessWriteable(op, "partition array") < 0) {
         return -1;
     }
@@ -1455,12 +1447,7 @@ PyArray_LexSort(PyObject *sort_keys, int axis)
         *((npy_intp *)(PyArray_DATA(ret))) = 0;
         goto finish;
     }
-    if (axis < 0) {
-        axis += nd;
-    }
-    if ((axis < 0) || (axis >= nd)) {
-        PyErr_Format(PyExc_ValueError,
-                "axis(=%d) out of bounds", axis);
+    if (check_and_adjust_axis(&axis, nd) < 0) {
         goto fail;
     }
 
diff --git a/numpy/core/src/multiarray/multiarraymodule.c b/numpy/core/src/multiarray/multiarraymodule.c
index f00de46c45aa..1c8d9b5e47e3 100644
--- a/numpy/core/src/multiarray/multiarraymodule.c
+++ b/numpy/core/src/multiarray/multiarraymodule.c
@@ -327,7 +327,6 @@ PyArray_ConcatenateArrays(int narrays, PyArrayObject **arrays, int axis)
     PyArray_Descr *dtype = NULL;
     PyArrayObject *ret = NULL;
     PyArrayObject_fields *sliding_view = NULL;
-    int orig_axis = axis;
 
     if (narrays <= 0) {
         PyErr_SetString(PyExc_ValueError,
@@ -345,13 +344,7 @@ PyArray_ConcatenateArrays(int narrays, PyArrayObject **arrays, int axis)
     }
 
     /* Handle standard Python negative indexing */
-    if (axis < 0) {
-        axis += ndim;
-    }
-
-    if (axis < 0 || axis >= ndim) {
-        PyErr_Format(PyExc_IndexError,
-                     "axis %d out of bounds [0, %d)", orig_axis, ndim);
+    if (check_and_adjust_axis(&axis, ndim) < 0) {
         return NULL;
     }
 
@@ -4109,6 +4102,24 @@ array_may_share_memory(PyObject *NPY_UNUSED(ignored), PyObject *args, PyObject *
     return array_shares_memory_impl(args, kwds, NPY_MAY_SHARE_BOUNDS, 0);
 }
 
+static PyObject *
+normalize_axis_index(PyObject *NPY_UNUSED(self), PyObject *args, PyObject *kwds)
+{
+    static char *kwlist[] = {"axis", "ndim", NULL};
+    int axis;
+    int ndim;
+
+    if (!PyArg_ParseTupleAndKeywords(args, kwds, "ii", kwlist,
+                                     &axis, &ndim)) {
+        return NULL;
+    }
+
+    if(check_and_adjust_axis(&axis, ndim) < 0) {
+        return NULL;
+    }
+
+    return PyInt_FromLong(axis);
+}
 
 static struct PyMethodDef array_module_methods[] = {
     {"_get_ndarray_c_version",
@@ -4284,6 +4295,8 @@ static struct PyMethodDef array_module_methods[] = {
         METH_VARARGS | METH_KEYWORDS, NULL},
     {"unpackbits", (PyCFunction)io_unpack,
         METH_VARARGS | METH_KEYWORDS, NULL},
+    {"normalize_axis_index", (PyCFunction)normalize_axis_index,
+        METH_VARARGS | METH_KEYWORDS, NULL},
     {NULL, NULL, 0, NULL}                /* sentinel */
 };
 
diff --git a/numpy/core/src/multiarray/shape.c b/numpy/core/src/multiarray/shape.c
index 3bee562be123..5207513bf531 100644
--- a/numpy/core/src/multiarray/shape.c
+++ b/numpy/core/src/multiarray/shape.c
@@ -705,12 +705,7 @@ PyArray_Transpose(PyArrayObject *ap, PyArray_Dims *permute)
         }
         for (i = 0; i < n; i++) {
             axis = axes[i];
-            if (axis < 0) {
-                axis = PyArray_NDIM(ap) + axis;
-            }
-            if (axis < 0 || axis >= PyArray_NDIM(ap)) {
-                PyErr_SetString(PyExc_ValueError,
-                                "invalid axis for this array");
+            if (check_and_adjust_axis(&axis, PyArray_NDIM(ap)) < 0) {
                 return NULL;
             }
             if (reverse_permutation[axis] != -1) {
diff --git a/numpy/core/src/umath/ufunc_object.c b/numpy/core/src/umath/ufunc_object.c
index 0bae2d591a84..af4ce12dbdf7 100644
--- a/numpy/core/src/umath/ufunc_object.c
+++ b/numpy/core/src/umath/ufunc_object.c
@@ -4036,12 +4036,7 @@ PyUFunc_GenericReduction(PyUFuncObject *ufunc, PyObject *args,
                 Py_DECREF(mp);
                 return NULL;
             }
-            if (axis < 0) {
-                axis += ndim;
-            }
-            if (axis < 0 || axis >= ndim) {
-                PyErr_SetString(PyExc_ValueError,
-                        "'axis' entry is out of bounds");
+            if (check_and_adjust_axis(&axis, ndim) < 0) {
                 Py_XDECREF(otype);
                 Py_DECREF(mp);
                 return NULL;
@@ -4058,18 +4053,11 @@ PyUFunc_GenericReduction(PyUFuncObject *ufunc, PyObject *args,
             Py_DECREF(mp);
             return NULL;
         }
-        if (axis < 0) {
-            axis += ndim;
-        }
         /* Special case letting axis={0 or -1} slip through for scalars */
         if (ndim == 0 && (axis == 0 || axis == -1)) {
             axis = 0;
         }
-        else if (axis < 0 || axis >= ndim) {
-            PyErr_SetString(PyExc_ValueError,
-                    "'axis' entry is out of bounds");
-            Py_XDECREF(otype);
-            Py_DECREF(mp);
+        else if (check_and_adjust_axis(&axis, ndim) < 0) {
             return NULL;
         }
         axes[0] = (int)axis;
diff --git a/numpy/core/tests/test_multiarray.py b/numpy/core/tests/test_multiarray.py
index 8229f1e1ab14..fa5051ba73fa 100644
--- a/numpy/core/tests/test_multiarray.py
+++ b/numpy/core/tests/test_multiarray.py
@@ -2013,13 +2013,13 @@ def test_partition(self):
             d = np.array([2, 1])
             d.partition(0, kind=k)
             assert_raises(ValueError, d.partition, 2)
-            assert_raises(ValueError, d.partition, 3, axis=1)
+            assert_raises(np.AxisError, d.partition, 3, axis=1)
             assert_raises(ValueError, np.partition, d, 2)
-            assert_raises(ValueError, np.partition, d, 2, axis=1)
+            assert_raises(np.AxisError, np.partition, d, 2, axis=1)
             assert_raises(ValueError, d.argpartition, 2)
-            assert_raises(ValueError, d.argpartition, 3, axis=1)
+            assert_raises(np.AxisError, d.argpartition, 3, axis=1)
             assert_raises(ValueError, np.argpartition, d, 2)
-            assert_raises(ValueError, np.argpartition, d, 2, axis=1)
+            assert_raises(np.AxisError, np.argpartition, d, 2, axis=1)
             d = np.arange(10).reshape((2, 5))
             d.partition(1, axis=0, kind=k)
             d.partition(4, axis=1, kind=k)
@@ -3522,8 +3522,8 @@ def test_object_argmin_with_NULLs(self):
 class TestMinMax(TestCase):
 
     def test_scalar(self):
-        assert_raises(ValueError, np.amax, 1, 1)
-        assert_raises(ValueError, np.amin, 1, 1)
+        assert_raises(np.AxisError, np.amax, 1, 1)
+        assert_raises(np.AxisError, np.amin, 1, 1)
 
         assert_equal(np.amax(1, axis=0), 1)
         assert_equal(np.amin(1, axis=0), 1)
@@ -3531,7 +3531,7 @@ def test_scalar(self):
         assert_equal(np.amin(1, axis=None), 1)
 
     def test_axis(self):
-        assert_raises(ValueError, np.amax, [1, 2, 3], 1000)
+        assert_raises(np.AxisError, np.amax, [1, 2, 3], 1000)
         assert_equal(np.amax([[1, 2, 3]], axis=1), 3)
 
     def test_datetime(self):
@@ -3793,7 +3793,7 @@ def test_object(self):  # gh-6312
 
     def test_invalid_axis(self): # gh-7528
         x = np.linspace(0., 1., 42*3).reshape(42, 3)
-        assert_raises(ValueError, np.lexsort, x, axis=2)
+        assert_raises(np.AxisError, np.lexsort, x, axis=2)
 
 class TestIO(object):
     """Test tofile, fromfile, tobytes, and fromstring"""
diff --git a/numpy/core/tests/test_numeric.py b/numpy/core/tests/test_numeric.py
index 4aa6bed3366a..906280e15353 100644
--- a/numpy/core/tests/test_numeric.py
+++ b/numpy/core/tests/test_numeric.py
@@ -1010,7 +1010,7 @@ def test_count_nonzero_axis(self):
 
         assert_raises(ValueError, np.count_nonzero, m, axis=(1, 1))
         assert_raises(TypeError, np.count_nonzero, m, axis='foo')
-        assert_raises(ValueError, np.count_nonzero, m, axis=3)
+        assert_raises(np.AxisError, np.count_nonzero, m, axis=3)
         assert_raises(TypeError, np.count_nonzero,
                       m, axis=np.array([[1], [2]]))
 
@@ -2323,10 +2323,10 @@ class TestRollaxis(TestCase):
 
     def test_exceptions(self):
         a = np.arange(1*2*3*4).reshape(1, 2, 3, 4)
-        assert_raises(ValueError, np.rollaxis, a, -5, 0)
-        assert_raises(ValueError, np.rollaxis, a, 0, -5)
-        assert_raises(ValueError, np.rollaxis, a, 4, 0)
-        assert_raises(ValueError, np.rollaxis, a, 0, 5)
+        assert_raises(np.AxisError, np.rollaxis, a, -5, 0)
+        assert_raises(np.AxisError, np.rollaxis, a, 0, -5)
+        assert_raises(np.AxisError, np.rollaxis, a, 4, 0)
+        assert_raises(np.AxisError, np.rollaxis, a, 0, 5)
 
     def test_results(self):
         a = np.arange(1*2*3*4).reshape(1, 2, 3, 4).copy()
@@ -2413,11 +2413,11 @@ def test_move_multiples(self):
 
     def test_errors(self):
         x = np.random.randn(1, 2, 3)
-        assert_raises_regex(ValueError, 'invalid axis .* `source`',
+        assert_raises_regex(np.AxisError, 'invalid axis .* `source`',
                             np.moveaxis, x, 3, 0)
-        assert_raises_regex(ValueError, 'invalid axis .* `source`',
+        assert_raises_regex(np.AxisError, 'invalid axis .* `source`',
                             np.moveaxis, x, -4, 0)
-        assert_raises_regex(ValueError, 'invalid axis .* `destination`',
+        assert_raises_regex(np.AxisError, 'invalid axis .* `destination`',
                             np.moveaxis, x, 0, 5)
         assert_raises_regex(ValueError, 'repeated axis in `source`',
                             np.moveaxis, x, [0, 0], [0, 1])
diff --git a/numpy/core/tests/test_shape_base.py b/numpy/core/tests/test_shape_base.py
index ac8dc1eea511..727608a175fa 100644
--- a/numpy/core/tests/test_shape_base.py
+++ b/numpy/core/tests/test_shape_base.py
@@ -184,8 +184,8 @@ def test_exceptions(self):
         for ndim in [1, 2, 3]:
             a = np.ones((1,)*ndim)
             np.concatenate((a, a), axis=0)  # OK
-            assert_raises(IndexError, np.concatenate, (a, a), axis=ndim)
-            assert_raises(IndexError, np.concatenate, (a, a), axis=-(ndim + 1))
+            assert_raises(np.AxisError, np.concatenate, (a, a), axis=ndim)
+            assert_raises(np.AxisError, np.concatenate, (a, a), axis=-(ndim + 1))
 
         # Scalars cannot be concatenated
         assert_raises(ValueError, concatenate, (0,))
@@ -294,8 +294,8 @@ def test_stack():
     expected_shapes = [(10, 3), (3, 10), (3, 10), (10, 3)]
     for axis, expected_shape in zip(axes, expected_shapes):
         assert_equal(np.stack(arrays, axis).shape, expected_shape)
-    assert_raises_regex(IndexError, 'out of bounds', stack, arrays, axis=2)
-    assert_raises_regex(IndexError, 'out of bounds', stack, arrays, axis=-3)
+    assert_raises_regex(np.AxisError, 'out of bounds', stack, arrays, axis=2)
+    assert_raises_regex(np.AxisError, 'out of bounds', stack, arrays, axis=-3)
     # all shapes for 2d input
     arrays = [np.random.randn(3, 4) for _ in range(10)]
     axes = [0, 1, 2, -1, -2, -3]
diff --git a/numpy/core/tests/test_ufunc.py b/numpy/core/tests/test_ufunc.py
index 3fea68700a5b..f7b66f90c0df 100644
--- a/numpy/core/tests/test_ufunc.py
+++ b/numpy/core/tests/test_ufunc.py
@@ -703,14 +703,14 @@ def test_zerosize_reduction(self):
 
     def test_axis_out_of_bounds(self):
         a = np.array([False, False])
-        assert_raises(ValueError, a.all, axis=1)
+        assert_raises(np.AxisError, a.all, axis=1)
         a = np.array([False, False])
-        assert_raises(ValueError, a.all, axis=-2)
+        assert_raises(np.AxisError, a.all, axis=-2)
 
         a = np.array([False, False])
-        assert_raises(ValueError, a.any, axis=1)
+        assert_raises(np.AxisError, a.any, axis=1)
         a = np.array([False, False])
-        assert_raises(ValueError, a.any, axis=-2)
+        assert_raises(np.AxisError, a.any, axis=-2)
 
     def test_scalar_reduction(self):
         # The functions 'sum', 'prod', etc allow specifying axis=0
diff --git a/numpy/lib/function_base.py b/numpy/lib/function_base.py
index ae1420b726a8..4d1ffbccc277 100644
--- a/numpy/lib/function_base.py
+++ b/numpy/lib/function_base.py
@@ -25,7 +25,7 @@
 from numpy.lib.twodim_base import diag
 from .utils import deprecate
 from numpy.core.multiarray import (
-    _insert, add_docstring, digitize, bincount,
+    _insert, add_docstring, digitize, bincount, normalize_axis_index,
     interp as compiled_interp, interp_complex as compiled_interp_complex
     )
 from numpy.core.umath import _add_newdoc_ufunc as add_newdoc_ufunc
@@ -4828,14 +4828,7 @@ def insert(arr, obj, values, axis=None):
             arr = arr.ravel()
         ndim = arr.ndim
         axis = ndim - 1
-    else:
-        if ndim > 0 and (axis < -ndim or axis >= ndim):
-            raise IndexError(
-                "axis %i is out of bounds for an array of "
-                "dimension %i" % (axis, ndim))
-        if (axis < 0):
-            axis += ndim
-    if (ndim == 0):
+    elif ndim == 0:
         # 2013-09-24, 1.9
         warnings.warn(
             "in the future the special handling of scalars will be removed "
@@ -4846,6 +4839,8 @@ def insert(arr, obj, values, axis=None):
             return wrap(arr)
         else:
             return arr
+    else:
+        axis = normalize_axis_index(axis, ndim)
     slobj = [slice(None)]*ndim
     N = arr.shape[axis]
     newshape = list(arr.shape)
diff --git a/numpy/lib/shape_base.py b/numpy/lib/shape_base.py
index 58e13533bf1f..62798286f32c 100644
--- a/numpy/lib/shape_base.py
+++ b/numpy/lib/shape_base.py
@@ -7,6 +7,7 @@
     asarray, zeros, outer, concatenate, isscalar, array, asanyarray
     )
 from numpy.core.fromnumeric import product, reshape, transpose
+from numpy.core.multiarray import normalize_axis_index
 from numpy.core import vstack, atleast_3d
 from numpy.lib.index_tricks import ndindex
 from numpy.matrixlib.defmatrix import matrix  # this raises all the right alarm bells
@@ -96,10 +97,7 @@ def apply_along_axis(func1d, axis, arr, *args, **kwargs):
     # handle negative axes
     arr = asanyarray(arr)
     nd = arr.ndim
-    if not (-nd <= axis < nd):
-        raise IndexError('axis {0} out of bounds [-{1}, {1})'.format(axis, nd))
-    if axis < 0:
-        axis += nd
+    axis = normalize_axis_index(axis, nd)
 
     # arr, with the iteration axis at the end
     in_dims = list(range(nd))
@@ -289,8 +287,7 @@ def expand_dims(a, axis):
     """
     a = asarray(a)
     shape = a.shape
-    if axis < 0:
-        axis = axis + len(shape) + 1
+    axis = normalize_axis_index(axis, a.ndim + 1)
     return a.reshape(shape[:axis] + (1,) + shape[axis:])
 
 row_stack = vstack
diff --git a/numpy/lib/tests/test_function_base.py b/numpy/lib/tests/test_function_base.py
index f69c24d59c5f..d914260add3c 100644
--- a/numpy/lib/tests/test_function_base.py
+++ b/numpy/lib/tests/test_function_base.py
@@ -466,8 +466,8 @@ def test_multidim(self):
                      insert(a, 1, a[:, 2,:], axis=1))
 
         # invalid axis value
-        assert_raises(IndexError, insert, a, 1, a[:, 2, :], axis=3)
-        assert_raises(IndexError, insert, a, 1, a[:, 2, :], axis=-4)
+        assert_raises(np.AxisError, insert, a, 1, a[:, 2, :], axis=3)
+        assert_raises(np.AxisError, insert, a, 1, a[:, 2, :], axis=-4)
 
         # negative axis value
         a = np.arange(24).reshape((2, 3, 4))
diff --git a/numpy/linalg/linalg.py b/numpy/linalg/linalg.py
index 03f456601954..119326912b96 100644
--- a/numpy/linalg/linalg.py
+++ b/numpy/linalg/linalg.py
@@ -25,6 +25,7 @@
     finfo, errstate, geterrobj, longdouble, rollaxis, amin, amax, product, abs,
     broadcast, atleast_2d, intp, asanyarray, isscalar, object_
     )
+from numpy.core.multiarray import normalize_axis_index
 from numpy.lib import triu, asfarray
 from numpy.linalg import lapack_lite, _umath_linalg
 from numpy.matrixlib.defmatrix import matrix_power
@@ -2225,13 +2226,8 @@ def norm(x, ord=None, axis=None, keepdims=False):
             return add.reduce(absx, axis=axis, keepdims=keepdims) ** (1.0 / ord)
     elif len(axis) == 2:
         row_axis, col_axis = axis
-        if row_axis < 0:
-            row_axis += nd
-        if col_axis < 0:
-            col_axis += nd
-        if not (0 <= row_axis < nd and 0 <= col_axis < nd):
-            raise ValueError('Invalid axis %r for an array with shape %r' %
-                             (axis, x.shape))
+        row_axis = normalize_axis_index(row_axis, nd)
+        col_axis = normalize_axis_index(col_axis, nd)
         if row_axis == col_axis:
             raise ValueError('Duplicate axes given.')
         if ord == 2:
diff --git a/numpy/linalg/tests/test_linalg.py b/numpy/linalg/tests/test_linalg.py
index a353271de9fd..b0a1f04d07e9 100644
--- a/numpy/linalg/tests/test_linalg.py
+++ b/numpy/linalg/tests/test_linalg.py
@@ -1102,8 +1102,8 @@ def test_bad_args(self):
             assert_raises(ValueError, norm, B, order, (1, 2))
 
         # Invalid axis
-        assert_raises(ValueError, norm, B, None, 3)
-        assert_raises(ValueError, norm, B, None, (2, 3))
+        assert_raises(np.AxisError, norm, B, None, 3)
+        assert_raises(np.AxisError, norm, B, None, (2, 3))
         assert_raises(ValueError, norm, B, None, (0, 1, 2))
 
 
diff --git a/numpy/ma/core.py b/numpy/ma/core.py
index a6f474b954c9..1b25725d1301 100644
--- a/numpy/ma/core.py
+++ b/numpy/ma/core.py
@@ -41,6 +41,7 @@
     getargspec, formatargspec, long, basestring, unicode, bytes, sixu
     )
 from numpy import expand_dims as n_expand_dims
+from numpy.core.multiarray import normalize_axis_index
 
 
 if sys.version_info[0] >= 3:
@@ -3902,7 +3903,9 @@ def __eq__(self, other):
                         axis = None
                     try:
                         mask = mask.view((bool_, len(self.dtype))).all(axis)
-                    except ValueError:
+                    except (ValueError, np.AxisError):
+                        # TODO: what error are we trying to catch here?
+                        #       invalid axis, or invalid view?
                         mask = np.all([[f[n].all() for n in mask.dtype.names]
                                        for f in mask], axis=axis)
                 check._mask = mask
@@ -3938,7 +3941,9 @@ def __ne__(self, other):
                         axis = None
                     try:
                         mask = mask.view((bool_, len(self.dtype))).all(axis)
-                    except ValueError:
+                    except (ValueError, np.AxisError):
+                        # TODO: what error are we trying to catch here?
+                        #       invalid axis, or invalid view?
                         mask = np.all([[f[n].all() for n in mask.dtype.names]
                                        for f in mask], axis=axis)
                 check._mask = mask
@@ -4340,7 +4345,7 @@ def count(self, axis=None, keepdims=np._NoValue):
 
             if self.shape is ():
                 if axis not in (None, 0):
-                    raise ValueError("'axis' entry is out of bounds")
+                    raise np.AxisError("'axis' entry is out of bounds")
                 return 1
             elif axis is None:
                 if kwargs.get('keepdims', False):
@@ -4348,11 +4353,9 @@ def count(self, axis=None, keepdims=np._NoValue):
                 return self.size
 
             axes = axis if isinstance(axis, tuple) else (axis,)
-            axes = tuple(a if a >= 0 else self.ndim + a for a in axes)
+            axes = tuple(normalize_axis_index(a, self.ndim) for a in axes)
             if len(axes) != len(set(axes)):
                 raise ValueError("duplicate value in 'axis'")
-            if builtins.any(a < 0 or a >= self.ndim for a in axes):
-                raise ValueError("'axis' entry is out of bounds")
             items = 1
             for ax in axes:
                 items *= self.shape[ax]
diff --git a/numpy/ma/extras.py b/numpy/ma/extras.py
index 29a15633d95c..7149b525bad1 100644
--- a/numpy/ma/extras.py
+++ b/numpy/ma/extras.py
@@ -36,6 +36,7 @@
 import numpy as np
 from numpy import ndarray, array as nxarray
 import numpy.core.umath as umath
+from numpy.core.multiarray import normalize_axis_index
 from numpy.lib.function_base import _ureduce
 from numpy.lib.index_tricks import AxisConcatenator
 
@@ -380,11 +381,7 @@ def apply_along_axis(func1d, axis, arr, *args, **kwargs):
     """
     arr = array(arr, copy=False, subok=True)
     nd = arr.ndim
-    if axis < 0:
-        axis += nd
-    if (axis >= nd):
-        raise ValueError("axis must be less than arr.ndim; axis=%d, rank=%d."
-            % (axis, nd))
+    axis = normalize_axis_index(axis, nd)
     ind = [0] * (nd - 1)
     i = np.zeros(nd, 'O')
     indlist = list(range(nd))
@@ -717,8 +714,8 @@ def _median(a, axis=None, out=None, overwrite_input=False):
 
     if axis is None:
         axis = 0
-    elif axis < 0:
-        axis += asorted.ndim
+    else:
+        axis = normalize_axis_index(axis, asorted.ndim)
 
     if asorted.ndim == 1:
         counts = count(asorted)
diff --git a/numpy/ma/tests/test_core.py b/numpy/ma/tests/test_core.py
index f72ddc5eace8..9d8002ed09e4 100644
--- a/numpy/ma/tests/test_core.py
+++ b/numpy/ma/tests/test_core.py
@@ -1030,7 +1030,7 @@ def test_count_func(self):
         res = count(ott, 0)
         assert_(isinstance(res, ndarray))
         assert_(res.dtype.type is np.intp)
-        assert_raises(ValueError, ott.count, axis=1)
+        assert_raises(np.AxisError, ott.count, axis=1)
 
     def test_minmax_func(self):
         # Tests minimum and maximum.
@@ -4409,7 +4409,7 @@ def test_count(self):
         assert_equal(count(a, axis=(0,1), keepdims=True), 4*ones((1,1,4)))
         assert_equal(count(a, axis=-2), 2*ones((2,4)))
         assert_raises(ValueError, count, a, axis=(1,1))
-        assert_raises(ValueError, count, a, axis=3)
+        assert_raises(np.AxisError, count, a, axis=3)
 
         # check the 'nomask' path
         a = np.ma.array(d, mask=nomask)
@@ -4423,13 +4423,13 @@ def test_count(self):
         assert_equal(count(a, axis=(0,1), keepdims=True), 6*ones((1,1,4)))
         assert_equal(count(a, axis=-2), 3*ones((2,4)))
         assert_raises(ValueError, count, a, axis=(1,1))
-        assert_raises(ValueError, count, a, axis=3)
+        assert_raises(np.AxisError, count, a, axis=3)
 
         # check the 'masked' singleton
         assert_equal(count(np.ma.masked), 0)
 
         # check 0-d arrays do not allow axis > 0
-        assert_raises(ValueError, count, np.ma.array(1), axis=1)
+        assert_raises(np.AxisError, count, np.ma.array(1), axis=1)
 
 
 class TestMaskedConstant(TestCase):
diff --git a/numpy/polynomial/chebyshev.py b/numpy/polynomial/chebyshev.py
index 3babb8fc2e95..49d0302e0e75 100644
--- a/numpy/polynomial/chebyshev.py
+++ b/numpy/polynomial/chebyshev.py
@@ -90,6 +90,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -936,10 +937,7 @@ def chebder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -1064,10 +1062,7 @@ def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py
index 0ebae2027792..a03fe722cb01 100644
--- a/numpy/polynomial/hermite.py
+++ b/numpy/polynomial/hermite.py
@@ -62,6 +62,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -700,10 +701,7 @@ def hermder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -822,10 +820,7 @@ def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
diff --git a/numpy/polynomial/hermite_e.py b/numpy/polynomial/hermite_e.py
index a09b66670a25..2a29d61cf6fd 100644
--- a/numpy/polynomial/hermite_e.py
+++ b/numpy/polynomial/hermite_e.py
@@ -62,6 +62,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -699,10 +700,7 @@ def hermeder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -821,10 +819,7 @@ def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
diff --git a/numpy/polynomial/laguerre.py b/numpy/polynomial/laguerre.py
index dfa997254eb8..c9e1302e133f 100644
--- a/numpy/polynomial/laguerre.py
+++ b/numpy/polynomial/laguerre.py
@@ -62,6 +62,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -697,10 +698,7 @@ def lagder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -822,10 +820,7 @@ def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py
index fdaa56e0c397..fa578360e435 100644
--- a/numpy/polynomial/legendre.py
+++ b/numpy/polynomial/legendre.py
@@ -86,6 +86,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -736,10 +737,7 @@ def legder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -864,10 +862,7 @@ def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
diff --git a/numpy/polynomial/polynomial.py b/numpy/polynomial/polynomial.py
index 19b085eaf20d..c357b48c9603 100644
--- a/numpy/polynomial/polynomial.py
+++ b/numpy/polynomial/polynomial.py
@@ -66,6 +66,7 @@
 import warnings
 import numpy as np
 import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
 
 from . import polyutils as pu
 from ._polybase import ABCPolyBase
@@ -540,10 +541,7 @@ def polyder(c, m=1, scl=1, axis=0):
         raise ValueError("The order of derivation must be non-negative")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c
@@ -658,10 +656,7 @@ def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
         raise ValueError("Too many integration constants")
     if iaxis != axis:
         raise ValueError("The axis must be integer")
-    if not -c.ndim <= iaxis < c.ndim:
-        raise ValueError("The axis is out of range")
-    if iaxis < 0:
-        iaxis += c.ndim
+    iaxis = normalize_axis_index(iaxis, c.ndim)
 
     if cnt == 0:
         return c