We overview recent existence results and techniques about KAM theory for PDEs.

We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on ${𝕋}^d,d\ge 1$, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty }$ then the solutions are $C^{\infty }$. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...