  Given a finite, flat and finitely presented group scheme $G$ over some base
$S$, we introduce the notion of ramified $G$-covers and study the moduli stack
$G$-Cov they form. The thesis is divided in three parts. The first one concerns
the case when $G$ is a diagonalizable group scheme and it essentially coincides
with arxiv:1106.2347. In the second part I deal with the general case. Assuming
that the base S is affine and given an $S$-scheme $T$, I interpret $G$-covers
of $T$ as particolar (lax) monoidal functors from the category of finite,
$G$-equivariant locally free sheaves over $S$ to the category of finite locally
free sheaves over $T$, extending the classical Tannakian correspondence between
$G$-torsors and strong monoidal functors as above. Using this point of view, I
prove that $G$-Cov is always reducible if $G$ is a non-abelian linearly
reductive group. When $G$ is constant and tame I also give a criterion to
detect when a $G$-cover of a regular in codimension one, integral scheme has
regular in codimension one total space in terms of the functor associated with
the cover. In the last part I focus on the case $G=S_3$, prove that $S_3$-Cov
has exactly two irreducible components and describe the principal one. I also
describe particular open loci of $S_3$-Cov, that is particular families of
$S_3$-covers, classify $S_3$-covers of regular schemes whose total space is
regular and compute the invariants of $S_3$-covers of smooth surfaces.
