We examine the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold. Our primary tool is the asymptotic expansion of the fundamental solution of the heat equation. Using terms in this expansion, the so-called "heat invariants," we show that the Laplace spectrum distinguishes elements within various classes of two-dimensional orbifolds. This is joint work with C. Gordon, S. Greenwald, D. Webb, and S. Zhu.