The cutoff phenomenon is a now classical topic in probability, that consists in proving an abrupt convergence to equilibrium for a sequence of Markovian dynamics. Classical examples include card shuffles, random walks on groups and graphs, or interacting particle systems. In this talk I will present an overview of the topic, and discuss some examples of cutoff for porous medium and fast diffusion equations in high dimension. Joint work with Djalil Chafaï and Nikita Simonov.