Partial compactifications of $\mathcal{M}_{g,n}$

These are geometrically defined intermediate subvarieties between $\mathcal{M}_{g,n}$ and $\overline{\mathcal{M}}_{g,n}$ whose topology and tautological rings may also have a nice structure. They include the moduli space $\mathcal{M}_{g,n}^{c}$ of curves of compact type (i.e. $\overline{\mathcal{M}}_{g,n} - \Delta_0$, or those curves whose dual graph is a tree), $\mathcal{M}_{g,n}^{\leq k}$ (curves with $\leq k$ rational components), and $\mathcal{M}_{g,n}^{\mathrm{rat}}$ (curves consisting of a single genus $g$ component attached to trees of rational tails).