Starting from basic principles, we will present some recent developments in the theory of rough paths and in the theory of sub-Riemannian diffusions. The first part of the talk will be devoted to the theory of rough paths. This theory was developed in the 1990s by T. Lyons, and allows to give a sense to solutions of differential equations driven by irregular paths. The theory itself has nothing to do with probability theory but has had a tremendous impact on several recent developments in stochastic analysis; it served as an inspiration to Hairer’s regularity structure theory, for which he was awarded the Fields medal in 2014. In the second part of the talk, we will address several problems in the geometric analysis of some sub-Riemannian manifolds, which can (surprisingly) be solved using diffusion semigroups techniques.