Mean field type models describing the limiting behavior of stochastic differential game problems as the number of players tends to infinity, have been recently introduced by J-M. Lasry and P-L. Lions. They may lead to systems of evolutive partial differential equations coupling a forward Bellman equation and a backward Fokker-Planck equation. The forward-backward structure is an important feature of this system, which makes it necessary to design new strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including: existence and uniqueness properties, a priori bounds on the solutions of the discrete schemes, convergence, and algorithms for solving the resulting nonlinear systems of equations. Some numerical experiments are presented. Collaborators: F. Camilli, I. Capuzzo Dolcetta, V. Perez