A natural family of random surfaces, which has received considerable attention recently in the context of JT gravity, is obtained from the Weil-Petersson measure on the moduli space of hyperbolic metrics on a genus-g surface with geodesic boundaries. In this talk I will describe how methods from random maps, i.e. random discrete surfaces, can be adapted to the hyperbolic world and give insight into the probabilistic aspects of the geometry of these hyperbolic surfaces when the number of boundaries becomes large. This talk is based on joint works with Nicolas Curien and with Thomas Meeusen and Bart Zonneveld.