In this talk, I will report on recent progress on wave turbulence, which is the turbulence observed in weakly nonlinear dispersive or wave equations. The main feature of wave turbulence is the existence of a kinetic equation describing energy transfers in phase space.
This field has developed very quickly over the last few years, along two main directions: on the one hand, the derivation of kinetic models from wave equations; and on the other hand, the analysis of the kinetic models. I will present some of these developments, emphasizing the rich interplay between the microscopic (wave / dispersive) and macroscopic (kinetic) descriptions.