Eukaryotic cell movement has essential functions ( in development, immunity or cancer) and so far very diverse cell migration patterns have been reported, but no general rule has emerged. We will show on the basis of experimental data in vitro and in vivo that in fact cell persistence, which quantifies the straightness of trajectories, is robustly coupled to cell migration speed. We suggest that this "universal" coupling constitutes a generic law of cell migration, which originates in the advection of polarity cues by an actin cytoskeleton undergoing flows at the cellular scale. The analysis relies on a theoretical model that yields a generic phase diagram of cellular trajectories, which recapitulates the range of observed migration patterns.