  Doped SrTiO$_3$, one of the most dilute bulk systems to display
superconductivity, is perhaps the first example of an unconventional
superconductor, as it does not fit into the standard BCS paradigm. More than
five decades of research has revealed a rich temperature-carrier concentration
phase diagram that showcases a superconducting dome, proximity to a putative
quantum critical point, Lifshitz transitions, a multi-gap pairing state and
unusual normal-state transport properties. Research has also extended beyond
bulk SrTiO$_3$, ushering the new field of SrTiO$_3$-based heterostructures.
Because many of these themes are also featured in other quantum materials of
contemporary interest, recent years have seen renewed interest in SrTiO$_3$.
Here, we review the challenges and recent progress in elucidating the
superconducting state of this model system. At the same time that its extreme
dilution requires to revisit several of the approximations that constitute the
successful Migdal-Eliashberg description of electron-phonon superconductivity,
including the suppression of the Coulomb repulsion via the
Tolmachev-Anderson-Morel mechanism, it opens interesting routes for alternative
pairing mechanisms whose applicability remains under debate. For instance,
pairing mechanisms involving longitudinal optical phonons have to overcome the
hurdles created by the anti-adiabatic nature of the pairing interaction,
whereas mechanisms that rely on the soft transverse optical phonons associated
with incipient ferroelectricity face challenges related to the nature of the
electron-phonon coupling. Proposals in which pairing is mediated by plasmons or
promoted locally by defects are also discussed. We finish by surveying the
existing evidence for multi-band superconductivity and outlining promising
directions that can potentially shed new light on the rich problem of
superconductivity in SrTiO$_3$.
