We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump Lévy processes with some σstable like behaviour. Included are σ -stable processes and fractional Laplace diffusion operators (− ) σ 2 , tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical

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... tions of MFG systems with nondegenerate diffusion operators of order σ ∈ (1, 2). We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs use pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques require no moment assumptions and use a weaker topology than Wasserstein.