Since its genesis in the early 1980s, the subject of quantum groups has grown very rapidly. Although much of the groundwork was laid by V. G. Drinfeld in his remarkable talk at the 1986 International Congress of Mathematicians in Berkeley, a number of basic issues in the theory were not resolved until later. In addition, important new developments occurred in the late 1980s and early 1990s, such as the crystal and canonical bases, and the applications of quantum groups in low-dimensional topology. By the late 1990s, however, the theory had reached a stage in which most of the foundational issues had been resolved, and many of the outstanding problems clearly formulated. It was felt that this was an opportune moment to hold a meeting of experts representing all the main strands of the theory, to take stock of what had been achieved so far and to discuss the most fruitful directions for future research. The result was the LMS Durham Symposium on Quantum Groups, which was held at Grey College in the University of Durham from 19 July to 29 July, 1999, and organised by S. Donkin, A. Pressley and A. Sudbery. The present volume is a record of some of the lectures given at the Symposium. Two lecture series are represented here which form excellent surveys of two important areas.