This talk (based on joint work with V.Delecroix, E.Goujard and P.Zograf) bridges certain aspects of flat and hyperbolic enumerative geometry. On the one hand, I will give a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of psi-classes (in the spirit of Mirzakhani's formula for Weil-Peterson volume of the moduli space of pointed curves). On the other hand, I will show that Mirzakhani's frequencies of simple closed hyperbolic geodesics of different combinatorial types coincide with the frequencies of the corresponding square-tiled surfaces. I will conclude with several conjectures concerning large genus asymptotics of intersection numbers of psi-classes and shape of a typical square-tiled surface of large genus.