This talk is devoted to the numerical approximation of mean field games problems. We consider two cases: a first order problem, i.e the diffusion is null, and a second order problem. For the first one, we propose aLagrange-Galerkin method to approximate the solution of a class of continuity equation, coupled with a semi-Lagrangian discretization of an Hamilton-Jacobi-Bellman equation, in order to obtain an approximation method for a first order Mean Field Games system. We prove a convergence result and we show some numerical simulations. For the second order case, we consider a Newton iterations approach for the continuous mean field game system, and we prove that the rate of convergence is quadratic. Finally we propose a semi Lagrangian scheme to approximate the continuous Newton iterations, and we show some numerical results. Joint work with Elisabetta Carlini Fabio Camilli and Francisco J. Silva.