The first main result is a probabilistic version of Thom transversality theorem - a necessary lubricant for the generic argument in Differential Topology (it is a more general version of Sard's theorem) - that allows to change the word "generic" to "almost every" in almost every situations.
Secondly, I will address the problem of describing the asymptotic behaviour of a sequence of Gaussian Random Fields. The prototypical example is a sequences of random (Kostlan) polynomials having {degree $\to\infty$}.