Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp`ere equations. This type of equations is precisely what is needed in order to construct K"ahler-Einstein metrics over irreducible singular K"ahler spaces with ample or trivial canonical sheaf and singular K"ahler-Einstein metrics over varieties of general type.