Ab initio quantum chemical methods can predict the equilibrium properties of small molecules in the gas phase to near-spectroscopic accuracy. However, a detailed understanding of a chemical reaction often requires knowledge of its dynamics, which in turn requires knowledge of the potential energy surface. The computation of potential energy surfaces to high accuracy is, unfortunately, much more challenging theoretically than the computation of equilibrium properties: if any bonds are formed or broken, then near degeneracies arise among the electron configurations, invalidating the assumptions of most standard methods. Although we focus on bond breaking (or, equivalently, bond formation), these same difficulties can occur for diradicals or metals of the first transition row. This chapter provides an introduction to the bond-breaking problem for nonspecialists and describes the reliability of different theoretical methods in such situations.