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.