In this talk I'll give an introduction to hyperbolic surfaces. Then I'll describe popular random and arithmetic models of them. I'll give an overview of the whole spectrum of the Laplacian on the surface, and then focus on
the potential spectral gap at zero. I'll explain both old and new results about the spectral gap
and finally highlight some remaining challenges.
Based on joint works with W. Hide, D. Puder, and R. van Handel.
