The purpose of this chapter is to introduce the notion of mean field game with a common noise. This terminology refers to the fact that in the finitely many player games from which the mean field game is derived, the states of the individual players are subject to correlated noise terms. In a typical model, each individual player feels an idiosyncratic noise as well as random shocks common to all the players. At the level of the mathematical analysis, the common noise introduces a randomization of most of the quantities and equations. In equilibrium, the statistical distribution of the population is no longer deterministic. One of the main feature of the chapter is the introduction and the analysis of the concepts of weak and strong solutions, very much in the spirit of the classical theory of stochastic differential equations.