Let (X,\omega) be a compact Hermitian manifold of complex dimension n. Consider the complex Monge-Ampère equation:
(\omega+dd^c\varphi)^n=cd\mu.
We prove the existence of continuous solutions and a stability result for measures well dominated by capacity. We also give a couple of characterizations of measures yielding Hölder continuous solutions of the Monge-Ampère equation. They allow us to identify certain classes of measures for which such solutions exist.
