It is well known that for close-to-integrable finite dimensional Hamiltonian systems "most" solutions live on maximal invariant tori, so it is very natural to wonder whether such a result can hold also in infinite dimensional systems. This is a very challenging problem and mostly still wide open. I will give an overview of recent results, main difficulties and open problems, concentrating on the case of the non-linear Schrödinger equation on the circle with a convolution potential.