Abstract: The theory of supercritical evolution PDEs remains mostly unknown in its basic questions (global existence, uniqueness, regularity). In this talk we present some results regarding these questions in the context of the two following models: the nonlinear Schrodinger equation (NLS), and the generalized SQG equations (gSQG). The proof of almost sure global well posedness relies on the Inviscid-Infinite dimensional limit (IID limit) approach. The IID limit can be seen as an independent method that relies on a combination of Bourgain's invariant measure argument and Kuksin's fluctuation-dissipation approach. Some parts of this work were carried out in collaboration with Xueying Yu and with Juraj Foldes.