Based on the Theory of Visual Attention (TVA; Bundesen, 1990, Psych Rev) we have developed a response time (RT) model for visual categorization. We propose that stimulus confusability leads to noise in the categorization process and that this noise can be countered by repeated categorization. Visual identifications are made conclusively in favor of one alternative over the other(s) once the number of tentative categorizations favoring this category exceeds the number favoring any other category by a critical amount (the response threshold). In a categorization task with only two alternatives, evidence accumulation follows a simple random walk with exponentially distributed interstep times. This Poisson random walk model is general enough to provide predictions for both correct and error RT distributions. The model is mathematically well tractable and allows for interesting generalizations, such as trial-to-trial variation in the Poisson processing rates and an extension of the random walk to n-alternatives. In an empirical test of the model we fitted the random walk model to data of a binary and a four-alternative orientation discrimination task. In both cases, the model predictions closely matched observed RT distributions. The agreement between theory and data is particularly remarkable in the multi-alternative case given the constraint that the model must explain RT distributions of correct and three different types of error responses simultaneously. The RT model inherits favorable properties of TVA such as well interpretable parameters of the visual identification process and the parameter estimates of the identification process resemble those obtained in classical TVA experiments.