In this talk we first review recent progress in the study of certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem) which stems from incorporating  to the deterministic toolbox, powerful but still classical tools from probability as well. We will explain some of these ideas and describe in more detail joint work with Gigliola Staffilani on the almost sure well-posedness for the periodic 3D quintic nonlinear Schrodinger equation in the supercritical regime; that is, below the critical space $H^1(\mathbb T^3)$. If time permits we will discuss non-deterministic  propagation of regularity for NLS in dimensions 1 and 2.