In this paper, we investigate the geometric properties of random hyperbolic surfaces of large genus. We describe the relationship
between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces. First, we study the asymptotic behavior of Weil-Petersson volume Vg,n of the moduli spaces of hyperbolic surfaces of genus
g with n punctures as g → ∞. Then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect
to the Weil-Petersson measure as g → ∞.