  In this paper we introduce a class of fully nonlinear mean field games posed
in $[0,T]\times\mathbb{R}^d$. We justify that they are related to controlled
local or nonlocal diffusions, and more generally in our setting, to a new
control interpretation involving time change rates of stochastic (L\'evy)
processes. The main results are existence and uniqueness of solutions under
general assumptions. These results are applied to strongly degenerate equations
of order less than one - and non-degenerate equations - including both local
second-order and nonlocal with fractional Laplacians. Uniqueness holds without
strict monotonicity of couplings or strict convexity of Hamiltonians, and in
the degenerate case, it relies on a new type of argument for (nonlocal)
Fokker-Planck equations. We consider a rich class of nonlocal operators and
processes and develop tools to work in the whole space without explicit moment
assumptions.
