We will study the spectrum of the adjacency and Laplacian operators of finite random graphs and Cayley graphs of groups. We first introduce their spectral measures and study their continuity properties with respect to the Benjamini-Schramm topology which we will define. We then explore the regularity properties of the spectral measure and put that into the framework of quantum percolation. The final part of the course is devoted to the convergence of extremal eigenvalues of random graphs.