In this talk, I'll describe the geometry of the moduli space $H_{g,n}$ of $n$-pointed, genus $g$ hyperelliptic curves. As $n$ grows relative to $g$, work of Casnati and Schwarz shows that $H_{g,n}$ goes from being rational (the simplest kind of variety) to general type (quite complicated). This suggests that we have hope of probing finer invariants of $H_{g,n}$ when $n$ is small relative to $g$. The Chow ring of $H_{g,n}$ is one such invariant. I will describe an inductive procedure for stratifying $H_{g,n}$ into nice pieces, which allows us to calculate its rational Chow ring when $n \leq 2g + 6$. This is joint work with Samir Canning.