The Laplacian matrix of a graph is the matrix L=D-A where D is the diagonal matrix of degrees and A is the adjacency matrix. The abelian group defined by L, is an important invariant of the graph. For example, its torsion subgroup, called the critical group, has order equal to the number of spanning trees if the graph is connected. The critical group also has an amusing interpretation in terms of a solitaire chip-firing game (or sandpile automaton) on a graph. In this talk, I will describe these connections and discuss how the critical group can be computed for various families of graphs, including the recent computation for Paley graphs (in joint work with Chandler and Xiang). using representation theory and number theory.