  We give a thorough study of Hurwitz stacks both in Galois and non galois
case. The construction is applied to revisit somme classical examples, the
stack of stable curves equipped with a level structure, and the stacks of
tamely ramified cyclic covers. We exhibit some tautological cohomology classes,
and give some universal relations between them. Applications are given to
Cornalba-Harris type relations, and to some Hurwitz-Hodge integrals.
