Lecture notes on the DiPerna-Lions theory in abstract measure spaces

These notes closely correspond to a series of lectures given by the first author in Toulouse, on the recent extension of the theory of ODE well-posedness to abstract spaces, jointly obtained by the two authors. In the last part, we describe some further developments with respect to the theory of (possibly degenerate) diffusion processes, in a similar setting, contained in the second author's PhD thesis.