This overview article reviews the structure of a fully statistical spoken dialogue system (SDS), using as illustration, various systems and components built at Cambridge over the last few years. Most of the components in an SDS are essentially classifiers which can be trained using supervised learning. However, the dialogue management component must track the state of the dialogue and optimise a reward accumulated over time. This requires techniques for statistical inference and policy optimisation using reinforcement learning. The potential advantages of a fully statistical SDS are the ability to train from data without hand-crafting, increased robustness to environmental noise and user uncertainty, and the ability to adapt and learn on-line.