Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve Y , a finite group G and a positive integer n we study smooth, proper families X → Y × S → S of Galois covers of Y with Galois group isomorphic to G branched in n points, parameterized by algebraic varieties S. When G is with trivial center we prove that the Hurwitz space HGn(Y ) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary G we prove that HGn (Y ) is a coarse moduli variety. For families of pointed Galois covers of (Y, y0) we prove that the Hurwitz space HGn (Y, y0) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group G. We use classical tools of algebraic topology and of complex algebraic geometry.