  An interesting question is whether two 3-manifolds can be distinguished by
computing and comparing their collections of finite covers; more precisely, by
the profinite completions of their fundamental groups. In this paper, we solve
this question completely for closed orientable Seifert fibre spaces. In
particular, all Seifert fibre spaces are distinguished from each other by their
profinite completions apart from some previously-known examples due to Hempel.
We also characterize when bounded Seifert fibre space groups have isomorphic
profinite completions, given some conditions on the boundary.
