  We study the Dirichlet problem for complex Monge-Ampere equations in
Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result
extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the
flat case. We also consider the equation on compact manifolds without boundary,
attempting to generalize Yau's theorems in the Kaehler case. As applications of
the main result we study some connections between the homogeneous complex
Monge-Ampere ({\em HCMA}) equation and totally real submanifolds, and a special
Dirichlet problem for the HCMA equation related to Donaldson's conjecture on
geodesics in the space of Kaehler metrics.
