Here, the Cauchy problems for linear and nonlinear nonlocal Schrödinger equa-tions are studied. The equation involves convolution integral operators with general kernel operator functions whose Fourier transforms are operator functions defined in a Banach space  together with some growth conditions. By assuming enough smoothness on the initial data and the operator functions, the local and global existence and uniqueness of solutions are established. We can obtain different classes of nonlocal Schrödinger equations, by choos-ing the space  and linear operators, which occur in a wide variety of physical systems.