In this video John Tate describes three of the seven $1,000,000 prize problems, announced by the Clay Institute. The problems discussed are: the Riemann Hypothesis, the Conjecture of Birch and Swinnerton-Dyer, and the P versus NP problem. These questions are among the most important unsolved problems in analysis, number theory, and theoretical computer science. John Tate's lecture gives a lucid description of the essential nature and history of these great problems. The lecture will be sure to captivate the imagination of anyone interested in the frontiers of mathematics.