Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as Masur-Veech volumes of strata, Siegel-Veech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.

After a general introduction to flat surfaces I will present a formula for the Masur-Veech volumes of principal strata of quadratic differentials in terms of intersection numbers involving psi-classes. This formula closely relates the counting of square-tiled surfaces with Mirzakhani's counting of simple closed geodesic multicurves on hyperbolic surfaces, and leads to several conjectures for the large genus asymptotics. This is a joint work with V.Delecroix, P.Zograf and A.Zorich.