A biased survey about random hyperbolic surfaces

I will survey recent developments concerning the geometric properties of random hyperbolic surfaces. While in low genus a clear connection with random planar maps has recently emerged, the geometry of random hyperbolic surfaces in large genus remains largely mysterious, despite significant progress on the spectral side. Starting from the breakthrough work of Maryam Mirzakhani on Weil–Petersson random surfaces, I will discuss what is known about typical geometric features of such surfaces.