This is the second in a series of talks on the seven "Millennium Prize Problems". The Clay Mathematics Institute has offered one million dollars for the solution to any one of these problems. In this talk, we will discuss the history and mathematical influence of the Poincaré conjecture. This remarkable conjecture, posed in 1904 by the great French mathematician, physicist and philosopher Henri Poincare, has had a profound impact on the development of modern topology. It remains wide open.