In the past ten years, an inductive procedure called topological recursion proved to provide a very efficient way of solving many problems of enumerative geometry through the computation of intersection of tautological classes over the Deligne-Mumford compactification of the moduli space of Riemann surfaces. In order to understand the mysterious geometric origin of this procedure, we developed a new inductive procedure called geometric recursion together with Andersen and Borot. This geometric recursion is a machinery defining all sorts of mapping class group invariant objects attached to surfaces. In this lecture, I will first review the original topological recursion formalism together with its application to the study of Cohomological Field Theories before presenting the formalism of geometric recursion. I will finally present some examples of application including some generalisation of Mirzakhani-McShane identities and the computation of some closed forms on Teichmuller space.