Social cognition depends on our capacity for ‘mentalizing’, or explaining an agent’s behaviour in terms of their mental states. The development and neural substrates of mentalizing are well-studied, but its computational basis is only beginning to be probed. Here we present a model of core mentalizing computations: inferring jointly an actor’s beliefs, desires and percepts from how they move in the local spatial environment. Our Bayesian theory of mind (BToM) model is based on probabilistically inverting artificial-intelligence approaches to rational planning and state estimation, which extend classical expected-utility agent models to sequential actions in complex, partially observable domains. The model accurately captures the quantitative mental-state judgements of human participants in two experiments, each varying multiple stimulus dimensions across a large number of stimuli. Comparative model fits with both simpler ‘lesioned’ BToM models and a family of simpler non-mentalistic motion features reveal the value contributed by each component of our model.