The Clay Mathematics Institute announced the Millennium Prize Problems to mark the 100th anniversary of Hilbert's lecture at the second ICM. These Problems were set forth by two of the greatest mathematicians of the 20th century: John Tate and Michael Atiyah. Atiyah's lecture discusses four of the Millennium Problems and places them in context. He discusses their relevance to a wide variety of mathematics in a style that is interesting to experts and accessible to the non-mathematician. The Problems described in this lecture are among the most important unsolved problems in branches of pure mathematics including analysis, geometry, topology, and mathematical physics. Solutions of these problems would also have profound implications for fluid dynamics and high energy physics. His lecture gives a lucid description of the essential nature and history of these great problems. The lecture which will be sure to captivate the imagination of anyone interested in the frontiers of mathematics.