We present the results of a detailed Molecular-dynamics study of single-atom diffusion on the surfaces Ag (100) and (111), and Au (111), using the embedded-atom method to describe the interactions between the atoms. We find that diffusion is Arrhenius-like up to temperatures corresponding to a large fraction of the activation energy. We demonstrate, in addition, that an excellent description of the rate of diffusion is provided by a simple transition-state theory, together with parameters that derive directly from the static potential-energy surface. The Model predicts very accurately the activation energies, while the prefactor for diffusion is obtained within a factor of 2, a discrepancy we attribute to the neglect, in the Model, of the details of the structure of the surface. At higher temperatures, diffusion becomes clearly non-Arrhenius, and the model fails.