Statistical methods have long been the dominant approach in speech recognition and probabilistic modelling in ASR is now a mature technology. The use of statistical methods in other areas of spoken dialogue is however more recent and rather less mature. This paper reviews spoken dialogue systems from a statistical modelling perspective. The complete system is first presented as a partially observable Markov decision process. The various sub-components are then exposed by introducing appropriate intermediate variables. Samples of existing work are reviewed within this framework, including dialogue control and optimisation, semantic interpretation, goal detection, natural language generation and synthesis.