Quantum Mechanics, the theory of matter on small scales, plays a centrally important role in many of the most important areas of science and technology (e.g. lasers, mesoscopic and nanoscopic systems). However, few quantum systems can be solved analytically. For the rest, methods of approximation are required. Among these, asymptotic methods based on classical (Newtonian) mechanics are of increasing importance, especially in mesoscopic and nanoscopic systems, which lie at the boundary between the classical and quantum worlds. Within classical mechanics there is a broad spectrum of qualitatively different dynamics, ranging from integrable (completely regular) to strongly chaotic (highly irregular). Quantum chaos is the area of research concerned with how this fact manifests itself in quantum mechanics. It is an exciting and rapidly developing field, encompassing the mathematical analysis of new quantum phenomena and a wide variety of applications in many areas of science and technology (e.g in nanoscale systems and microlasers). There are deep connections with Random Matrix Theory - the study of the statistical distribution of the eigenvalues of matrices picked at random from some suitably defined ensemble - Ergodic Theory, and several areas of Number Theory, such as the theory of the Riemann zeta function and other related objects. Many fundamental developments in the subject have followed from work carried out here in Bristol. There is a close relationship with the Dynamical Systems and Quantum Information areas, and with the group in Physics led by Prof Sir Michael Berry FRS. There is also a close connection with the Number Theory and Ergodic Theory research areas.