The (extremely simple) Gronwall estimate allows to prove uniqueness of the solution of ODEs and some regularity of the solution with respect to the initial point. This in turns implies that the advection of regular profiles through regular velocities stays regular. In this colloquium I will explain in which (very strong) sense the Gronwall estimate is optimal. I will also describe some intriguing connections with mixing problems in fluid dynamics. This is based on ongoing research in collaboration with G. Alberti (Pisa) and A. Mazzucato (Penn State).