  In this article we classify quadruple Galois canonical covers of smooth
surfaces of minimal degree. The classification shows that they are either
non-simple cyclic covers or bi-double covers.
  If they are bi-double then they are all fiber products of double covers. We
construct examples to show that all the possibilities in the classification do
exist. There are implications of this classification that include the existence
of families with unbounded geometric genus and families with unbounded
irregularity, in sharp contrast with the case of double and triple canonical
covers. Together with the results of Horikawa and Konno for double and triple
covers, a pattern emerges that motivates some general questions on the
existence of higher degree canonical covers, some of which are answered in this
article.
