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.