Billiards are widely studied dynamical systems with many open problems. In the talk we will consider the question when the billiard flow on a convex body is continuous and see how this leads to a certain class of metric spaces, namely Riemannian orbifolds that I will introduce. For instance, a convex polytope has a continuous billiard flow if and only if it is a Riemannian orbifold. Moreover, I will interpret this question in the more general setting of Alexandrov geometry and explain how this viewpoint helps to answer it for convex bodies without corners. The latter in particular relates the billiard flow of a smooth convex body to the geodesic flow on its boundary. On the way, we will also encounter some open questions and other related results. No prior knowledge on Riemannian orbifolds or Alexandrov geometry is assumed.