  We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for
the complex Monge-Am\`ere equation on a general Hermitian manifold with
non-empty boundary. We prove optimal subsolution theorems: for bounded and
H\"older continuous quasi-plurisubharmonic functions. The continuity of the
solution is proved for measures that well dominated by capacity, for example
measures with $L^p$, $p>1$ densities, or moderate measures in the sense of
Dinh-Nguyen-Sibony.
