In this talk we will discuss a number of techniques to establish existence of smooth solutions of a class of mean-field games which have a variational structure. We start by showing that a number of mean-field games with local dependence on the player's density can be regarded as Euler-Lagrange equations for certain functionals. These functionals are convex but in a large number of important examples are not coercive. We will then discuss a number of techniques to establish a-priori estimates which when coupled with continuation methods allow to prove the existence of smooth solutions.