  We study complex Monge-Ampere equations on Hermitian manifolds, extending
classical existence results of Yau and Aubin in the Kahler case, and those of
Caffarelli, Kohn, Nirenberg and Spruck for the Dirichlet problem in $C^n$. As
an application we generalize existing results on the Donaldson conjecture on
geodesics in the space of Kahler metrics to the Hermitian setting.
