The application of random-matrix theory (RMT) to compound-nucleus (CN) reactions is reviewed. An introduction into the basic concepts of nuclear scattering theory is followed by a survey of phenomenological approaches to CN scattering. The implementation of a random-matrix approach into scattering theory leads to a statistical theory of CN reactions. Since RMT applies generically to chaotic quantum systems, that theory is, at the same time, a generic theory of quantum chaotic scattering. It uses a minimum of input parameters (average S-matrix and mean level spacing of the CN). Predictions of the theory are derived with the help of field-theoretical methods adapted from condensed-matter physics and compared with those of phenomenological approaches. Thorough tests of the theory are reviewed, as are applications in nuclear physics, with special attention given to violation of symmetries (isospin, parity) and time-reversal invariance.PACS numbers: * Electronic address: mitchell@tunl.duke.edu † Electronic address: richter@ikp.tu-darmstadt.de ‡ Electronic address: Hans.Weidenmueller@mpi-hd.mpg.de problem. Motivated by the fundamental interest in CN scattering and by the need in other fields of physics (neutron physics, shielding problems, nuclear astrophysics, etc.) to have a viable theory of CN reactions with predictive power, that work was carried on for a number of years and led to partial insights into the connection between RMT and CN scattering.