  Mean Field Games (MFG) have been introduced to tackle games with a large
number of competing players. Considering the limit when the number of players
is infinite, Nash equilibria are studied by considering the interaction of a
typical player with the population's distribution. The situation in which the
players cooperate corresponds to Mean Field Control (MFC) problems, which can
also be viewed as optimal control problems driven by a McKean-Vlasov dynamics.
These two types of problems have found a wide range of potential applications,
for which numerical methods play a key role since most models do not have
analytical solutions. In these notes, we review several aspects of numerical
methods for MFG and MFC. We start by presenting some heuristics in a basic
linear-quadratic setting. We then discuss numerical schemes for
forward-backward systems of partial differential equations (PDEs), optimization
techniques for variational problems driven by a Kolmogorov-Fokker-Planck PDE,
an approach based on a monotone operator viewpoint, and stochastic methods
relying on machine learning tools.
