diff --git a/INSTALL.rst b/INSTALL.rst
index b8d940971513..2c1e93d3b018 100644
--- a/INSTALL.rst
+++ b/INSTALL.rst
@@ -275,7 +275,7 @@ not needed during the wheel install.
 ::
 
   # create a new environment with the required packages
-  conda create -n "matplotlib_build" python=3.5 numpy python-dateutil pyparsing pytz tornado cycler tk libpng zlib freetype
+  conda create -n "matplotlib_build" python=3.7 numpy python-dateutil pyparsing pytz tornado cycler tk libpng zlib freetype
   activate matplotlib_build
   # if you want a qt backend, you also have to install pyqt (be aware that pyqt doesn't mix well if
   # you have created the environment with conda-forge already activated...)
diff --git a/build_alllocal.cmd b/build_alllocal.cmd
index 54bc69432fb7..11bf74e7c298 100644
--- a/build_alllocal.cmd
+++ b/build_alllocal.cmd
@@ -1,6 +1,6 @@
 :: This assumes you have installed all the dependencies via conda packages:
 :: # create a new environment with the required packages
-:: conda create -n "matplotlib_build" python=3.5 numpy python-dateutil pyparsing tornado cycler tk libpng zlib freetype
+:: conda create -n "matplotlib_build" python=3.7 numpy python-dateutil pyparsing tornado cycler tk libpng zlib freetype
 :: activate matplotlib_build
 :: if you want qt backend, you also have to install pyqt
 :: conda install pyqt
diff --git a/doc/devel/testing.rst b/doc/devel/testing.rst
index b4eb4ded7af9..651bbba8a8a1 100644
--- a/doc/devel/testing.rst
+++ b/doc/devel/testing.rst
@@ -238,16 +238,12 @@ example <https://travis-ci.org/msabramo/matplotlib>`_.
 Using tox
 ---------
 
-`Tox <https://tox.readthedocs.io/en/latest/>`_ is a tool for running
-tests against
-multiple Python environments, including multiple versions of Python
-(e.g., 3.5, 3.6) and even different Python implementations
-altogether (e.g., CPython, PyPy, Jython, etc.)
-
-Testing all versions of Python (3.5, 3.6, ...) requires
-having multiple versions of Python installed on your system and on the
-PATH. Depending on your operating system, you may want to use your
-package manager (such as apt-get, yum or MacPorts) to do this.
+`Tox <https://tox.readthedocs.io/en/latest/>`_ is a tool for running tests
+against multiple Python environments, including multiple versions of Python
+(e.g., 3.6, 3.7) and even different Python implementations altogether
+(e.g., CPython, PyPy, Jython, etc.), as long as all these versions are
+available on your system's $PATH (consider using your system package manager,
+e.g. apt-get, yum, or Homebrew, to install them).
 
 tox makes it easy to determine if your working copy introduced any
 regressions before submitting a pull request. Here's how to use it:
diff --git a/lib/matplotlib/font_manager.py b/lib/matplotlib/font_manager.py
index 135e202167cf..b3bf781422ae 100644
--- a/lib/matplotlib/font_manager.py
+++ b/lib/matplotlib/font_manager.py
@@ -191,9 +191,8 @@ def win32InstalledFonts(directory=None, fontext='ttf'):
                     direc = direc.split("\0", 1)[0]
                     try:
                         path = Path(directory, direc).resolve()
-                    except (FileNotFoundError, RuntimeError):
-                        # Don't fail with invalid entries (FileNotFoundError is
-                        # only necessary on Py3.5).
+                    except RuntimeError:
+                        # Don't fail with invalid entries.
                         continue
                     if path.suffix.lower() in fontext:
                         items.add(str(path))
diff --git a/setup.py b/setup.py
index e0e90dd46075..f339807cc56b 100644
--- a/setup.py
+++ b/setup.py
@@ -11,10 +11,9 @@
 from setuptools.command.test import test as TestCommand
 from setuptools.command.build_ext import build_ext as BuildExtCommand
 
-if sys.version_info < (3, 5):
+if sys.version_info < (3, 6):
     error = """
-Matplotlib 3.0+ does not support Python 2.x, 3.0, 3.1, 3.2, 3.3, or 3.4.
-Beginning with Matplotlib 3.0, Python 3.5 and above is required.
+Beginning with Matplotlib 3.1, Python 3.6 or above is required.
 
 This may be due to an out of date pip.
 
@@ -82,7 +81,6 @@
     'License :: OSI Approved :: Python Software Foundation License',
     'Programming Language :: Python',
     'Programming Language :: Python :: 3',
-    'Programming Language :: Python :: 3.5',
     'Programming Language :: Python :: 3.6',
     'Programming Language :: Python :: 3.7',
     'Topic :: Scientific/Engineering :: Visualization',
@@ -232,7 +230,7 @@ def build_extensions(self):
         classifiers=classifiers,
         download_url="http://matplotlib.org/users/installing.html",
 
-        python_requires='>=3.5',
+        python_requires='>=3.6',
         # List third-party Python packages that we require
         install_requires=install_requires,
         setup_requires=setup_requires,
diff --git a/tutorials/introductory/customizing.py b/tutorials/introductory/customizing.py
index 09a6312c0e29..9b7ae96f2641 100644
--- a/tutorials/introductory/customizing.py
+++ b/tutorials/introductory/customizing.py
@@ -154,8 +154,8 @@
 #
 # 4. :file:`{INSTALL}/matplotlib/mpl-data/matplotlibrc`, where
 #    :file:`{INSTALL}` is something like
-#    :file:`/usr/lib/python3.5/site-packages` on Linux, and maybe
-#    :file:`C:\\Python35\\Lib\\site-packages` on Windows. Every time you
+#    :file:`/usr/lib/python3.7/site-packages` on Linux, and maybe
+#    :file:`C:\\Python37\\Lib\\site-packages` on Windows. Every time you
 #    install matplotlib, this file will be overwritten, so if you want
 #    your customizations to be saved, please move this file to your
 #    user-specific matplotlib directory.