By analyzing the dynamics of model potential energy surfaces we systematically investigate the processes involved in passing from a high energy state to the global minimum and how the probability of reaching the global minimum depends upon the topography and topology of the potential energy surface (PES). Relaxation to the global minimum is easiest for PES’s consisting of a single funnel (a set of convergent pathways which lead to the global minimum) with low barriers and a significant potential energy gradient towards the global minimum. The presence of additional funnels on the surface can severely reduce the rate of relaxation to the global minimum. Such secondary funnels act most efficiently as kinetic traps when they terminate at a low energy minimum, have a steep potential energy gradient and are wide (i.e., have a large configurational entropy) compared to the primary funnel. Indeed, it is even possible to construct PES’s for which the system relaxes to the minimum at the bottom of a secondary funnel rather than the global minimum and then remains in this metastable state over a long time scale. Our results for these model PES’s are discussed in the context of theoretical and experimental knowledge of the dynamics of proteins, clusters, and glasses.