The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as the failure of materials, earthquakes, global warming, demographic patterns, and financial crises. In this talk, Didier Sornette describes a simple, powerful, and general theory of how, why, and when stock markets crash. Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, into an accelerating rise of the market price, otherwise known as a "bubble." This view implies the possibility of predicting such events and Sornette will describe the current status of predictions that he and his collaborators have made for events in various markets.