  We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective
line having only one ramified point over each branch point. We start with the
case of genus-0 covers, using a combination of limit linear series theory and
group theory to show that these spaces are always irreducible. In the case of
four branch points, we also compute the associated Hurwitz numbers. Finally, we
give a conditional result in the higher-genus case, requiring at least 3g
simply branched points. These results have equivalent formulations in group
theory, and in this setting complement results of Conway-Fried-Parker-Volklein.
