**Abstract : ** Consider a (possibly time-dependent) vector field $v$ on the Euclidean space.
The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$
is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$
and solving the ODE $\dot{\gamma} (t) = v (t, \gamma (t))$. The theorem looses its validity as soon as $v$ is slightly less regular.
However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory put forward by DiPerna and Lions
in the 80es 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 {\em 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 $h$-principle.