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Abstract:  Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales. Despite the fact that wave fields in nature are enormously diverse, there is a common mathematical concept that can be used to describe the processes of random wave interactions: the wave kinetic equations. After the production of the first Bose-Einstein Condensates (BECs), there has been an explosion of research on the kinetic theory associated to BECs and their thermal clouds. In this talk, we will summarize our recent progress on this topic.