  While there is much work and many conjectures surrounding the intersection
theory of the moduli space of curves, relatively little is known about the
intersection theory of the Hurwitz space $\mathcal{H}_{k, g}$ parametrizing
smooth degree $k$, genus $g$ covers of $\mathbb{P}^1$. Let $k = 3, 4, 5$. We
prove that the rational Chow rings of $\mathcal{H}_{k,g}$ stabilize in a
suitable sense as $g$ tends to infinity. In the case $k = 3$, we completely
determine the Chow rings for all $g$. We also prove that the rational Chow
groups of the simply branched Hurwitz space $\mathcal{H}^s_{k,g} \subset
\mathcal{H}_{k,g}$ are zero in codimension up to roughly $g/k$. In subsequent
work, results developed in this paper are used to prove that the Chow rings of
$\mathcal{M}_7, \mathcal{M}_8,$ and $\mathcal{M}_9$ are tautological.
