  In this paper, we prove a uniform and sharp estimate for the modulus of
continuity of solutions to complex Monge-Amp\`ere equations, using the
PDE-based approach developed by the first three authors in their approach to
supremum estimates for fully non-linear equations in K\"ahler geometry. As an
application, we derive a uniform diameter bound for K\"ahler metrics satisfying
certain Monge-Amp\`ere equations.
