A family of random matrices converges strongly if, in addition to the convergence of their joint moments, the operator norm of any noncommutative polynomial in these matrices converges to that of its limiting free probability model. We will survey this notion, from its original motivation in operator algebras and the first breakthrough of Haagerup and Thorbjørnsen for Gaussian random matrices, to the remarkable acceleration of the last decade, with consequences ranging from operator algebras to spectral gaps of random graphs. Along the way, we will discuss contributions by Bordenave and the speaker, Parraud, Bandeira, Boedihardjo and van Handel, and Chen, Garza-Vargas, Tropp and van Handel, and conclude with recent joint work with Yamagishi. The talk is intended for a broad mathematical audience.