We consider the universal family $E_n^d$ of superelliptic curves: each curve $��_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $��_n^d\hookrightarrow E_n^d\to C_n$ 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�� 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 $��_n^d,$ 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 $d=2.$

39 pages, 7 figures, 5 tables, improved results, section 9 and some references added. arXiv admin note: text overlap with arXiv:1708.00207