  We show that Rashba spin-orbit interaction (RSOI) modifies electron-electron
interaction vertex giving rise to a spectrum of novel phenomena. First, the
spin-orbit-modified Coulomb interactions induce $p$-wave superconducting order,
without any need for other mediators of attraction. Remarkably, two distinct
superconducting phases arise in 3D systems, mirroring the $\mathrm{A}$ or
$\mathrm{B}$ phases of $^3\mathrm{He}$, depending on the sign of the SOI
constant. In contrast, 2D systems exhibit $p^x\pm i p^y$ order parameter,
leading to time-reversal-invariant topological superconductivity. Second, a
sufficiently strong RSOI induces ferromagnetic ordering. It is associated with
a deformation of the Fermi surface, which may lead to a Lifshitz transition
from a spherical to a toroidal Fermi surface, with a number of experimentally
observable signatures. Finally, in sufficiently clean Rashba materials,
ferromagnetism and $p$-wave superconductivity may coexist. This state resembles
the $\mathrm{A}_1$ phase of $^3\mathrm{He}$, yet it may avoid nodal points due
to the toroidal shape of the Fermi surface.
