  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 $\mathscr{H}_{k, g}$ parametrizing
smooth degree $k$, genus $g$ covers of $\mathbb{P}^1$. Let $k = 3, 4, 5$. We
prove that the rational Chow rings of $\mathscr{H}_{k,g}$ stabilize in a
suitable sense as $g$ tends to infinity. In the case $k = 3$, we completely
determine the Chow rings for all $g$. In codimension $1$, our results hold
integrally, thus determining the integral Picard groups
$\mathrm{Pic}(\mathscr{H}_{k,g})$. We also prove that the rational Chow groups
of the simply branched Hurwitz space $\mathscr{H}^s_{k,g} \subset
\mathscr{H}_{k,g}$ 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
$\mathscr{M}_7, \mathscr{M}_8,$ and $\mathscr{M}_9$ are tautological.
