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Wave (or weak) turbulence theory aims at describing the turbulent regime occurring in weakly nonlinear wave and dispersive equations. The central object of the theory is a kinetic equation, known as the kinetic wave equation, which describes energy transfers in phase space in the aforementioned regime. While this theory is widely used by physicists, not much was known rigorously until recently. But the mathematics of wave turbulence has become a very active field, of which I will try to give an overview.