  SrTiO$_3$ is a unique example of a system which exhibits both quantum
paraelectricity and superconductivity. Thus, it is expected that the
superconducting state is closely related to the intrinsic ferroelectric
instability. Indeed, recent experiments suggest existence of a coexistent phase
of superconductivity and ferroelectricity in Ca-substituted SrTiO$_3$. In this
paper, we propose that SrTiO$_3$ can be a platform of the ferroelectric
superconductivity, which is characterized by a ferroelectric transition in the
superconducting state. By analyzing a multiorbital model for $t_{2g}$
electrons, we show that the ferroelectric superconductivity is stabilized
through two different mechanisms which rely on the presence of the spin-orbit
coupling. First, the ferroelectric superconducting state is stabilized in the
dilute carrier density regime due to a ferroelectricity-induced Lifshitz
transition. Second, it is stabilized under a magnetic field independent of the
carrier density. The importance of the multiorbital or multiband nature for the
ferroelectric superconductivity is clarified. Then, we predict a topological
Weyl superconducting state in the ferroelectric superconducting phase of
SrTiO$_3$.
