  Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer
$n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study
smooth, proper families, parameterized by algebraic varieties, of pointed
degree $d$ covers of $(Y,y_0)$, $(X,x_{0})\to (Y,y_0)$, branched in $n$ points
of $Y\setminus y_{0}$, whose monodromy group equals $G$. We construct a Hurwitz
space $H$, an algebraic variety whose points are in bijective correspondence
with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We
construct explicitly a family parameterized by $H$, whose fibers belong to the
corresponding equivalence classes, and prove that it is universal. We use
classical tools of algebraic topology and of complex algebraic geometry.
