A mean-field game is a game with a continuum of players,  describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. We first study the asymptotic behavior of Nash equilibria in static games with a large number of agents. In particular, we establish law of large number limits and large deviation principles for the set of Nash equilibria and discuss applications to congestion games and the price of anarchy. Then we discuss stochastic differential games, which are often understood via the so-called "master equation", which is an infinite-dimensional PDE for the value function. We will show how analysis of sufficiently smooth solutions to the master equation play a role in analyzing large deviation principles for mean-field games. This is based on joint works with Francois Delarue and Daniel Lacker.