  In this manuscript we construct global in time classical solutions to mean
field games master equations in the lack of idiosyncratic noise in the
individual agents' dynamics. These include both deterministic models and
dynamics driven solely by a Brownian common noise. We consider a general class
of non-separable Hamiltonians and final data functions that are supposed to be
displacement monotone. Our main results unify and generalize in particular some
of the well-posedness results on displacement monotone master equations
obtained recently by Gangbo--M\'esz\'aros and Gangbo--M\'esz\'aros--Mou--Zhang.
