This talk will focus on a new strategy introduced in the last few years which has turned out to be a very useful tool to study the asymptotic behaviour of random matrices and compute matrix integrals. It relies on the theory of free stochastic calculus developed by Biane and Speicher in the late '90s. I will explain the heuristic behind those results, as well as list a few of them. In particular I will explain how it was used to deduce a proof of a conjecture on the strong convergence of a family of random matrices which implies the Peterson–Thom conjecture thanks to the work of Ben Hayes.