From the earliest times, the geometry of space has been intimately involved with physics. As science has evolved and our understanding has deepened, the relations between geometry and physics have become subtler and more complex. At the time of this lecture, fundamentally new ideas from both areas were dramatically altering conceptions about the nature of the universe. In this presentation, Sir Michael Atiyah, one of the foremost mathematicians of the 20th century, discusses some of the deep connections that have been discovered between mathematics and quantum physics. Starting with the viewpoints of Euclid and Newton, Atiyah moves on to ideas growing out of Jones' work on knots in 3-space and Donaldson's work on 4-manifolds. In describing how Witten has brought these developments into contact with quantum field theory, Atiyah shows how quantum field theory is in itself an effort to understand the structure of a vacuum. A witty, engaging, and clear-sighted lecturer, Atiyah makes this fascinating topic accessible to audiences with a general scientific background.