  The aim of this paper is to prove that a large class of quaternionic slice
regular functions result to be (ramified) covering maps. By means of the
topological implications of this fact and by providing further topological
structures, we are able to give suitable natural conditions for the existence
of $k$-th $\star$-roots of a slice regular function. Moreover, we are also able
to compute all the solutions which, quite surprisingly, in the most general
case, are in number of $k^2$. The last part is devoted to compute the monodromy
and to present a technique to compute all the $k^2$ roots starting from one of
them.
