The Kuramoto model is the archetype of heterogeneous systems of (globally) coupled oscillators with dissipative dynamics. In this model, the order parameter that quantifies the population synchrony decays to 0 in time, as long as the interaction strength remains small (so that the uniformly distributed stationary solution remains stable). While this phenomenon has been identified since the first studies of the model, its proof remained to be provided (most studies in the literature are limited to the linearized dynamics). The goal of this talk is to address this issue. I will present rigorous results on the Kuramoto dynamics, and in particular, I will sketch a proof of nonlinear damping of the order parameter in the incoherent phase. Joint work with D. Gérard-Varet and G. Giacomin