  Materials with charged chiral quasiparticles in external parallel electric
and magnetic fields can support an electric current that grows linearly in
time, corresponding to diverging DC conductivity. From experimental viewpoint,
this "Chiral Magnetic Superconductivity" (CMS) is thus analogous to
conventional superconductivity. However the underlying physics is entirely
different -- the CMS does not require a condensate of Cooper pairs breaking the
gauge degeneracy, and is thus not accompanied by Meissner effect. Instead, it
owes its existence to the (temperature-independent) quantum chiral anomaly and
the conservation of chirality. As a result, this phenomenon can be expected to
survive to much higher temperatures. Even though the chirality of
quasiparticles is not strictly conserved in real materials, the chiral magnetic
superconductivity should still exhibit itself in AC measurements at frequencies
larger than the chirality-flipping rate, and in microstructures of Dirac and
Weyl semimetals with thickness below the mean chirality-flipping length that is
about 1-100 $\mu$m. In nuclear physics, the CMS should contribute to the
charge-dependent elliptic flow in heavy ion collisions.
