After having introduced some definitions, we will present a classification holding true in an open set of the total space of such dynamical systems. The phenomena that occur comprise the foliation of the phase space in smooth KAM tori, in tori of lower regularity down to simply measurable ones, and (for systems where the tori break down completely) Weak Mixing in the fibres and Unique Ergodicity in the space of Distributions. This classification is based on works of H. Eliasson, R. Krikorian and the author, among others.