  Let $\Omega\subseteq M$ be a bounded domain with a smooth boundary
$\partial\Omega$, where $(M,J,g)$ is a compact, almost Hermitian manifold. The
main result of this paper is to consider the Dirichlet problem for a complex
Monge-Amp\`{e}re equation on $\Omega$. Under the existence of a $C^{2}$-smooth
strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve
this Dirichlet problem. Our method is based on the properties of subsolutions
which have been widely used for fully nonlinear elliptic equations over
Hermitian manifolds.
