  We consider the universal family $E_n^d$ of superelliptic curves: each curve
$\Sigma_n^d$ in the family is a $d$-fold covering of the unit disk, totally
ramified over a set $P$ of $n$ distinct points; $\Sigma_n^d\hookrightarrow
E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$
distinct points. We find that $E_n^d$ is the classifying space for the complex
braid group of type $B(d,d,n)$ and we compute a big part of the integral
homology of $E_n^d,$ including a complete calculation of the stable groups over
finite fields by means of Poincar\`e series. The computation of the main part
of the above homology reduces to the computation of the homology of the
classical braid group with coefficients in the first homology group of
$\Sigma_n^d,$ endowed with the monodromy action. While giving a geometric
description of such monodromy of the above bundle, we introduce generalized
$1\over d$-twists, associated to each standard generator of the braid group,
which reduce to standard Dehn twists for $d=2.$
