Mathematical modeling of fuel cells is necessary in order to understand the precise effect of operating conditions and material properties on the performance of fuel cells. This article presents a rigorous mathematical model to predict performance of proton exchange membrane (PEM) fuel cells of arbitrary three-dimensional geometry. The model is based on solution of the conservation equations of mass, momentum, energy, species and electric current in a fully integrated finite-volume solver. Simulations are performed in two-dimensional and three-dimensional geometries assuming that no liquid water is formed. Results are compared against experimental data under various operating conditions, and show that three-dimensional modeling is key to predicting performance of PEM fuel cells.