abstract: We will start by presenting two basic probabilistic effects for questions concerning the regularity of functions and nonlinear operations on functions. We will then overview well-posedenss results for the nonlinear wave equation, the nonlinear Schroedinger equation and the nonlinear heat equation in the presence of singular randomness. In the remaining part of the lectures, we will focus on the nonlinear wave equation with initial data distributed according to non degenerate gaussian measures on Sobolev spaces of varying regularity. We will introduce the notation of super-ctitical regularity by presenting the relevant ill-posedness results. We will then prove probabilistic well-posedness for data of super-critical regularity, by constructing the dynamics almost surely with respect to the corresponding gaussian measure. We will then turn to the transport properties of our gaussian measures under the flow of the non linear wave equation. We will finally discuss the proof of a very resent result constructing the relevant dynamics when the measures are supported by Sobolev spaces of negative indexes.