Eigenvalues of the hyperbolic Laplacian and Random Matrix Theory

I will discuss some of the interactions between number theory and the spectral theory of the Laplacian. Some have very classical background, such as the connection with lattice point problems. Others are newer, including connections with Random Matrix Theory, the zeros of the Riemann zeta function, and the work of Maryam Mirzakhani on the moduli space of hyperbolic surfaces.