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We prove existence and uniqueness of classical solutions of the masterequation for mean field game (MFG) systems with fractional and nonlocaldiffusions. We cover a large class of L\'evy diffusions of order greater thanone, including purely nonlocal, local, and even mixed local-nonlocal operators.In the process we prove refined well-posedness results for the MFG systems,results that include the mixed local-nonlocal case. We also show variousauxiliary results on viscous Hamilton-Jacobi equations, linear parabolicequations, and linear forward-backward systems that may be of independentinterest. This includes a rigorous treatment of certain equations and systemswith data and solutions in the duals of H\"older spaces $C^\gamma_b$ on thewhole of $\mathbb{R}^d$. We do not assume existence of any moments for theinitial distributions of players. In a future work we will use the results ofthis paper to prove the convergence of $N$-player games to mean field games as$N\to\infty$.

The master equation for mean field game systems with fractional and nonlocal diffusions