We present a computational framework for understanding Theory of Mind (ToM): the human capacity for reasoning about agents' mental states such as beliefs and desires. Our Bayesian model of ToM (or BToM) expresses the predictive model of belief- and desire-dependent action at the heart of ToM as a partially observable Markov decision process (POMDP), and reconstructs an agent's joint belief state and reward function using Bayesian inference, conditioned on observations of the agent's behavior in some environmental context. We test BToM by showing participants sequences of agents moving in simple spatial scenarios and asking for joint inferences about the agents' desires and beliefs about unobserved aspects of the environment. BToM performs substantially better than two simpler variants: one in which desires are inferred without reference to an agent's beliefs, and another in which beliefs are inferred without reference to the agent's dynamic observations in the environment.