We introduce the notion of Galois cover for a finite group and discuss
the problems of constructing them and of the
geometry of the stack -Cov they form. When is abelian, we describe certain families of
-covers in terms of combinatorial data associated with . In the
general case, we present a correspondence between -covers and
particular monoidal functors and study the problem of Galois covers of
normal varieties whose total space is normal.