  In this paper, we prove a $C^{1,1}$ estimate for solutions of complex
Monge-Amp\`{e}re equations on compact almost Hermitian manifolds. Using this
$C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate
Monge-Amp\`{e}re equations, the corresponding Dirichlet problems and the
singular Monge-Amp\`{e}re equations. We also study the singularities of the
pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$
estimate is valid for a kind of complex Monge-Amp\`{e}re type equations. As a
geometric application, we prove the $C^{1,1}$ regularity of geodesics in the
space of Sasakian metrics.
