In 2000, the Clay Foundation of Cambridge, Massachusetts, announced a historic competition: whoever could solve any of seven extraodinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive million in prize money. The solutions, if any, to the so-called Milliennium Problems would play a strong role in determining the course of mathematics in the 21st century. They encompass many of the most fascinating areas of pure and applied mathematics, from topology and number theory to particle physics, cryptography, computing and even aircraft design. Keith Devlin describes here what the seven problems are, how they came about, and what they mean for mathematics and science. In the hands of Devlin, each Millennium Problem becomes a window onto the deepest questions in the field.