The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin glasses, or interacting particle systems. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then present a new approach based on entropy and curvature.