In the talk I will disuss the existence of the geodesic flow and the invariance of the Liouville measure in non-smooth settings. A central role will play the so-called metric-measure boundary, an object from the geometric measure theory, which detects the boundary of a Riemannian manifold in the smooth setting and controls the average non-flatness of the space in more general situations. The talk will be based on a joint work with Vitaly Kapovitch and Anton Petrunin and an ongoing project with Daniele Semola and Stephan Stadler.