Stacks of ramified Galois cover

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