  Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite
type and $x\in X(S)$ a section. The aim of the present paper is to establish
the existence of the fundamental group scheme of $X$, when $X$ has reduced
fibers or when $X$ is normal. We also prove the existence of a group scheme,
that we will call the quasi-finite fundamental group scheme of $X$ at $x$,
which classifies all the quasi-finite torsors over $X$, pointed over $x$. We
define Galois torsors, which play in this context a role similar to the one of
Galois covers in the theory of \'etale fundamental group.
