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Given a smooth, projective curve $Y$, a finite group $G$ and a positiveinteger $n$ we study smooth, proper families $X\to Y\times S\to S$ of Galoiscovers of $Y$ with Galois group isomorphic to $G$ branched in $n$ points,parameterized by algebraic varieties $S$. When $G$ is with trivial center weprove that the Hurwitz space $H^G_n(Y)$ is a fine moduli variety for thiscategory and construct explicitly the universal family. For arbitrary $G$ weprove that $H^G_n(Y)$ is a coarse moduli variety. For families of pointedGalois covers of $(Y,y_0)$ we prove that the Hurwitz space $H^G_n(Y,y_0)$ is afine moduli variety, and construct explicitly the universal family, forarbitrary group $G$. We use classical tools of algebraic topology and ofcomplex algebraic geometry.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve