  We review the generalization of field theory to space-time with noncommuting
coordinates, starting with the basics and covering most of the active
directions of research. Such theories are now known to emerge from limits of M
theory and string theory, and to describe quantum Hall states. In the last few
years they have been studied intensively, and many qualitatively new phenomena
have been discovered, both on the classical and quantum level.
  To appear in Reviews of Modern Physics.
