Mean field games are systems of nonlinear PDE which have been introduced as limits of problems from game theory (as the number of agents tends to infinity). The system consists of a coupled forward parabolic and backward parabolic equation. Our viewpoint is that together, the forward and backward parabolic equations form a quasilinear elliptic system. Another elliptic system, arising in fluid dynamics, is the classical vortex sheet problem, which describes the motion of an interface between two fluids. Using insights gained from the analysis of vortex sheets, we prove an existence theorem for mean field games.