The author, noting that basic facts about circulant matrices and its relationship to the Discrete Fourier Transform were rediscovered over and over again, summarized these facts in 1979. Circulant matrices have since have since played an increasingly large role in applications and algebraists, numerical analysts, combinatorialists and physicists have pushed forward the development of generalized circulants. Such matrices are now often seen as special instances of structured or patterned matrices. The outgrowth of the simple notion of a circulant matrix has therefore been both vast and profound. Readers who are interested in applications or generalizations of circulants beyond what is given in this volume may also find a list of publications (and their bibliographies) to be of use.