Statistical dialogue systems are motivated by the need for a data-driven framework that reduces the cost of laboriously hand-crafting complex dialogue managers and that provides robustness against the errors created by speech recognisers operating in noisy environments. By including an explicit Bayesian model of uncertainty and by optimising the policy via a reward-driven process, partially observable Markov decision processes (POMDPs) provide such a framework. However, exact model representation and optimisation is computationally intractable. Hence, the practical application of POMDP-based systems requires efficient algorithms and carefully constructed approximations. This review article provides an overview of the current state of the art in the development of POMDP-based spoken dialogue systems.