The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define a quadratic form. The main purpose of the talk is to raise this question: "What can quadratic forms tell us about graphs?" As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. No knowledge of quadratic forms will be assumed, but the essential definitions and facts will be sketched.