Optimal transport is a very versatile theory that lifts the geometry on a given underlying space to the overlying Wasserstein space of probability measures. Based on this rich interpretation, Jordan, Kinderlehrer and Otto showed 25 years ago that the (linear!) heat equation can be seen as the (highly nonlinear!) Wasserstein gradient flow of the Boltzmann entropy in the space of probability measures, thus providing a strong form of the second principle of thermodynamics. This sparked tremendous interest due to the interconnection between various mathematical branches, ranging from applied PDEs, numerical analysis, probability and interacting particle systems, metric geometry, functional analysis, and more. In this talk I will try to review the main ideas behind this formalism, and show how classical optimal transport can be extended to study a broad class of evolution equations from a variational standpoint (eg. Hele-Shaw dynamics, reaction-diffusion, multiphase flows, evolutionary genetics, etc.).