These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students.

The equations which govern the motion of two bodies moving under each other's gravitational influence have **lots** of symmetry: although they constitute a 12th-order system of **nonlinear** ordinary differential equations, the solutions turn out to be simple ellipses and hyperbolae. Add a third body, however, and some of this symmetry is lost. Just how much is lost depends on how close the third body is to the other two; while two bodies follow predictable paths, many three-body systems follow chaotic paths.

Emmy Noether showed that physical symmetries such as rotational invariance (the outcome of an experiment isolated from the rest of the universe does not depend on its orientation) always have associated with them ``integrals of the motion'' - combinations of dependent variables which don't vary in time. The study of such symmetries is fundamental to all of modern theoretical physics. I will discuss how these ideas can help us uncover some of the secrets of the three-body problem and I will present a brand new integral of the general three-body problem.