In these lectures, we review a series of optimal transport (OT) problems of growing complexity. Surprisingly enough, in this seemingly narrow framework, we will encounter nonlinear PDEs of very different type, such as the Monge–Ampère équation, the Euler equations of incompressible fluids, the hydrostatic Boussinesq equations in convection theory, the Born–Infeld equation of electromagnetism, showing the hidden richness of the concept of optimal transportion