Motivated by a few concrete examples such as bird flocking, congestion for crowd motion, and bee swarming, we introduce the mean field game paradigm; and we present the underpinnings of the original analytic approach based on partial differential equations, and the probabilistic approach based on forward-backward stochastic differential equations. The theoretical results presented in the first part of the talk will be illustrated by the results of numerical implementations of a mean field game model for the synchronization of circadian rhythms and a mathematical model for jet lag recovery.