Appears in collection : Algèbre, Géométrie et Physique : une conférence en l'honneur

I will try to describe how dynamics of certain natural flows on surfaces is governed by dynamics of the Teichmuller flow on the moduli space and by geometry of the Hodge bundle. In particular, I will try to present recent results of J. Athreya, A. Eskin, M. Mirzakhani, M. Kontsevich, and myself making the relation between the two problems rather explicit. As a model case I will consider billiards in the plane with periodic polygonal obstacles (developing ideas of V. Delecroix, P. Hubert, and S. Lelièvre).