Quasi-plurisubharmonic envelopes 3: Solving Monge-Amp`ere equations on hermitian manifolds

We develop a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kahler geometry. In cite{GL21b} we have studied the behavior of Monge-Amp`ere volumes on hermitian manifolds. We extend here the techniques of cite{GL21a} to the hermitian setting and use the bounds established in cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp`ere equations on compact hermitian manifolds.