Two new integrable nonlocal Davey–Stewartson equations are introduced. These equations provide two-spatial dimensional analogues of the integrable, nonlocal nonlinear Schrö-dinger equation introduced in Ablowitz and Musslimani (2013 Phys. Rev. Lett. 110 064105). Furthermore, like the latter equation, they also possess a PT symmetry and, as it is well known, this symmetry is important for the occurence of such equations in nonlinear optics. A method for solving the initial value problem of these integrable equations is discussed. It is shown that the technique used for constructing these novel integrable equations has general validity; as an illustrative example, an additional two-dimensional integrable generalization of the nonlocal nonlinear Schrödinger is also presented.