Well posedness of ODE's for nonsmooth velocities and in non Euclidean ambient spaces: a survey

In this lecture I will make a survey of the theory of regular lagrangian flows, that allows to prove existence, uniqueness and stability for ODE's associated to vector fields with little regularity, beyond the classical Cauchy-Lipschitz theory. Initiated in a minal work by Di Perna and Lions who dealt with Sobolev vector fields, the RLF axiomatization provides natural links with Probability. More recent developments, including the regularity of the flow map, counterexamples and the case when even the ambient space is not smooth, will be discussed.