We discuss the long time average of Mean Field Games systems as the time horizon tends to infinity and the convergence towards a stationary ergodic mean field game, both in case of local and nonlocal coupling in the cost functional. We also prove that convergence holds at exponential rate, exploiting two completely different approaches; in case of a local, strongly monotone, coupling, we use estimates coming from the Hamiltonian structure of the system, while in case of a nonlocal, regularizing coupling, we use smoothing properties and the exponential decay of the linearized system. Joint works with P. Cardaliaguet, J-M. Lasry and P-L. Lions.