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In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomologyring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? Thisquestion has been fully answered in genus $0$ by Keel (the Chow and cohomologyrings are tautological for all $n$) and genus $1$ by Belorousski (the rings aretautological if and only if $n \leq 10$). For $g \geq 2$, work of van Zelmshows the Chow and cohomology rings are not tautological once $2g + n \geq 24$,leaving finitely many open cases. Here, we prove that the Chow and cohomologyrings of $\overline{\mathcal{M}}_{g,n}$ are isomorphic and generated bytautological classes for $g = 2$ and $n \leq 9$ and for $3 \leq g \leq 7$ and$2g + n \leq 14$. For such $(g, n)$, this implies that the tautological ring isGorenstein and $\overline{\mathcal{M}}_{g,n}$ has polynomial point count.

On the Chow and cohomology rings of moduli spaces of stable curves