In this talk I shall report recent progress on continuous-time behavioural portfolio choice under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility functions and probability distortions. It is shown that the model well-posedness becomes a prominent issue in such a behavioural model. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. Time permitting I will also discuss on the incomplete market and single-period models as well as the equity premium puzzle.