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In this paper, we prove a $C^{1,1}$ estimate for solutions of complexMonge-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 degenerateMonge-Amp\`{e}re equations, the corresponding Dirichlet problems and thesingular Monge-Amp\`{e}re equations. We also study the singularities of thepluricomplex 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 ageometric application, we prove the $C^{1,1}$ regularity of geodesics in thespace of Sasakian metrics.

$C^{1,1}$ regularity of degenerate complex Monge-Amp\`{e}re equations and some applications