The canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift-diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift-diffusion process to multiple alternatives. The competition between n alternatives takes place in a linear subspace of [Formula: see text] dimensions; that is, there are [Formula: see text] decision variables, which are coupled through correlated noise sources. I derive the multiple-alternative DDM starting from a system of coupled, linear firing rate equations. I also show that a Bayesian sequential probability ratio test for multiple alternatives is, in fact, equivalent to these same linear DDMs, but with time-varying thresholds. If the original neuronal system is nonlinear, one can once again derive a model describing a lower-dimensional diffusion process. The dynamics of the nonlinear DDM can be recast as the motion of a particle on a potential, the general form of which is given analytically for an arbitrary number of alternatives.