Tautological rings

The tautological ring $R^*(\overline{\mathcal{M}}_{g,n})$ is the subring of the the Chow ring $A^*(\overline{\mathcal{M}}_{g,n})$ which is meant to contain all of the natural geometric information. Faber and Pandharipande gave the following elegant formulation (which is equivalent to previous definitions). There are forgetful morphisms $\overline{\mathcal{M}}_{g,n} \to \overline{\mathcal{M}}_{g,n-1}$ and gluing morphisms $\overline{\mathcal{M}}_{g,n+1}\times \overline{\mathcal{M}}_{h,m+1} \to \overline{\mathcal{M}}_{g+h,n+m}$ and $\overline{\mathcal{M}}_{g,n+2} \to \overline{\mathcal{M}}_{g+1,n}$. The system of tautological rings is then the smallest system of $\mathbb{Q}$-subalgebras of the Chow rings which is closed under the gluing and pushforward maps and which contains all of the Witten classes $\psi_i$. Tautological rings for the uncompactified moduli space and its partial compactifications are defined by restriction.