  The theory of mean field games aims at studying deterministic or stochastic
differential games (Nash equilibria) as the number of agents tends to infinity.
Since very few mean field games have explicit or semi-explicit solutions,
numerical simulations play a crucial role in obtaining quantitative information
from this class of models. They may lead to systems of evolutive partial
differential equations coupling a backward Bellman equation and a forward
Fokker-Planck equation. In the present survey, we focus on such systems. The
forward-backward structure is an important feature of this system, which makes
it necessary to design unusual 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 convergence, variational aspects and algorithms for
solving the resulting systems of nonlinear equations. Finally, we discuss in
details two applications of mean field games to the study of crowd motion and
to macroeconomics, a comparison with mean field type control, and present
numerical simulations.
