  On a compact Riemannian manifold, we study the various dynamical properties
of the Schr\"odinger flow $(e^{it\Delta/2})$, through the notion of
semiclassical measures and the quantum-classical correspondence between the
Schr\"odinger equation and the geodesic flow. More precisely, we are interested
in its high-frequency behavior, as well as its regularizing and unique
continuation-type properties. We survey a variety of results illustrating the
difference between positive, negative and vanishing curvature.
