  In this article we shall give an account of certain developments in knot
theory which followed upon the discovery of the Jones polynomial in 1984. The
focus of our account will be recent glimmerings of understanding of the
topological meaning of the new invariants. A second theme will be the central
role that braid theory has played in the subject. A third will be the unifying
principles provided by representations of simple Lie algebras and their
universal enveloping algebras. These choices in emphasis are our own. They
represent, at best, particular aspects of the far-reaching ramifications that
followed the discovery of the Jones polynomial.
