Artin introduced braid groups into mathematical literature in 1925. In the years since, and particularly in the last five to ten years, braid groups have played diverse and unexpected roles in widely different areas of mathematics, including knot theory, homotopy theory, singularity theory, and dynamical systems. Most recently, the area of operator algebras has brought striking new applications to knots and links. This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group, held at the University of California, Santa Cruz, in July 1986. This interdisciplinary conference brought together leading specialists in diverse areas of mathematics to discuss their discoveries and to exchange ideas and problems concerning this important and fundamental group.Because the proceedings present a mix of expository articles and new research, this volume will be of interest to graduate students and researchers who wish to learn more about braids, as well as more experienced workers in this area. The required background includes the basics of knot theory, group theory, and low-dimensional topology.