Consider a (possibly time-dependent) vector field ** V** on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindeloef) Theorem states that, if the vector field

**is Lipschitz in space, for every initial datum**

*V***there is a unique trajectory**

*x***starting at**

*f***at time**

*x***and solving the ODE**

*t=0**(*

**f**'*)*

**t***=*(

**V***(*

**t**,**f***)). The theorem looses its validity as soon as*

**t***is slightly less regular. However, if we bundle all trajectories into a global map allowing*

**V****to vary, a celebrated theory put forward by DiPerna and Lions in the 1980's show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for**

*x**almost every*initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov's

**-principle.**

*h*