  Recent experiments on Nb-doped SrTiO$_3$ have shown that the superconducting
energy gap to the transition temperature ratio maintains the
Bardeen-Cooper-Schrieffer (BCS) value throughout its superconducting dome.
Motivated by these and related studies, we show that the Cooper pairing
mediated by a single soft transverse-optical phonon is the most natural
mechanism for such a superconducting dome given experimental constraints, and
present the microscopic theory for this pairing mechanism. Furthermore, we show
that this mechanism is consistent with the $T^2$ resistivity in the normal
state. Lastly, we discuss what physical insights SrTiO$_3$ provides for
superconductivity in other quantum paraelectrics such as KTaO$_3$.

Superconductivity