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.
Our goal, throughout this review, is to present the material in the most transparent and nontechnical manner possible in order to help readers who work in other areas to learn as much as possible about the state of the art in knot theory. Thus, when we give "proofs", they will be, at best, sketches of proofs. We hope there will be enough detail so that, say, a diligent graduate student who is motivated to read a little beyond this paper will be able to fill in the gaps.