In this talk we address the numerical approximation of Mean Field Games (MFG) with local couplings. For finite difference discretizations of the MFG system, we follow a variational approach and show that the discretised MFG can be obtained as the optimality system of a convex optimization problem. We study different proximal-type methods for convex, non-smooth optimization. These algorithms have several interesting features, such as global convergence, mass preservation, and stability with respect to the viscosity parameter. We present numerical experiments assessing the performance of the proposed methods, and discuss extensions related to gradient flows and mean field control. In collaboration with L. Briceño-Arias (UTFSM, CL) and F. J. Silva (Univ. Limoges, FR).