We discuss models of updating that depart from Bayes’ rule even when it is well-defined. After reviewing Bayes’ rule and its foundations, we begin our analysis with models of non-Bayesian behavior arising from a bias, a pull toward suboptimal behavior due to a heuristic or a mistake. Next, we explore deviations caused by individuals questioning the prior probabilities they initially used. We then consider non-Bayesian behavior resulting from the optimal response to constraints on perception, cognition, or memory, as well as models based on motivated beliefs or distance minimization. Finally, we briefly discuss models of updating after zero probability events.