  These theories, which are surely some of the simplest possible quantum field
theories, were introduced in a paper of Dijkgraaf and Witten. The path integral
reduces to a finite sum, so it is quite amenable to direct mathematical study.
Although the theory exisits in arbitrary dimensions, it is most interesting in
$2+1$~dimensions, where it has a ``modular structure.'' This is related to
quantum groups, and the precise details may give clues as to what happens in
other contexts.
  This paper is written using AMSTeX 2.1, which can be obtained via ftp from
the American Mathematical Society (instructions included). 1 encapsulated
postscript file was submitted separately in uuencoded tar-compressed format.
