  We prove existence and uniqueness of classical solutions of the master
equation for mean field game (MFG) systems with fractional and nonlocal
diffusions. We cover a large class of L\'evy diffusions of order greater than
one, 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 various
auxiliary results on viscous Hamilton-Jacobi equations, linear parabolic
equations, and linear forward-backward systems that may be of independent
interest. This includes a rigorous treatment of certain equations and systems
with data and solutions in the duals of H\"older spaces $C^\gamma_b$ on the
whole of $\mathbb{R}^d$. We do not assume existence of any moments for the
initial distributions of players. In a future work we will use the results of
this paper to prove the convergence of $N$-player games to mean field games as
$N\to\infty$.
