Structure, Composition, and High-Field Superconductivity in Metal-Rich η-Carbide-Type Compounds

$\mathrm{\eta}$-Carbide-type compounds have recently emerged as a diverse class of materials in the study of superconductivity. These phases contribute to a growing family of metal-rich quantum materials that exhibit unusual superconducting properties emerging from complex metallic bonding. Several members of the $\mathrm{\eta}$-carbide-type phases have been found to be bulk superconductors -- such as Nb$_4$Rh$_2$C$_{1-\delta}$, Ta$_4$Rh$_2$C$_{1-\delta}$, Ti$_4$Ir$_2$O$_{1-\delta}$, and Ti$_4$Co$_2$O$_{1-\delta}$ -- with transition temperatures up to $T_{\rm c} \approx$ 10 K and upper critical fields as high as $\mu_0 H_{\rm c2}(0) \approx$ 30 T. Whereas the transition temperatures may fall within the range typical for intermetallic superconductors, the pronounced violation of the weak-coupling Pauli limit in many of these crystallographically high-symmetry materials is noteworthy. Here, we review recent progress on superconducting $\mathrm{\eta}$-carbide-type phases, emphasizing how crystal symmetry, synthetic challenges, transition-metal composition, and electronic structure govern their superconducting properties. Furthermore, we outline open questions and future directions, including the possible discovery of new $\mathrm{\eta}$-carbide-type materials.