The gravitational N-body problem dominates much of theoretical astrophysics. It arises in problems ranging from the motion of artificial and natural satellites to the behaviour of stars in star clusters and galaxies. In the past, research in Celestial Mechanics has focused on analytical and numerical studies of ordered systems, with chaotic behaviour or disorder being the fascinating anomaly. Recently, however, scientists have begun to forge the link between order and disorder: designing new methods to recognise chaos' signature and developing new theories on the sources of chaotic behaviour. They are applying these techniques successfully to solve real-world problems in solar system dynamics, galactic dynamics and other areas of physics such as particle acceleration.Based on the recent NATO Advanced Study Institute "Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems", this state of the art textbook, written by internationally renowned experts, provides an invaluable reference volume for all students and researchers in gravitational n-body systems. The contributions are especially designed to give a systematic development from the fundamental mathematics which underpin modern studies of ordered and chaotic behaviour in n-body dynamics to their application to real motion in planetary systems. This volume presents an up-to-date synoptic view of the subject.