  Chirality of neutrinos modifies the conventional kinetic theory and
hydrodynamics, leading to unusual chiral transport related to quantum anomalies
in field theory. We argue that these corrections have new phenomenological
consequences for hot and dense neutrino gases, especially in core-collapse
supernovae. We find that the neutrino density can be converted to the fluid
helicity through the chiral vortical effect. This fluid helicity effectively
acts as a chiral chemical potential for electrons via the momentum exchange
with neutrinos and induces a "helical plasma instability" that generates a
strong helical magnetic field. This provides a new mechanism for converting the
gravitational energy released by the core collapse to the electromagnetic
energy and potentially explains the origin of magnetars. The other possible
applications of the neutrino chiral transport theory are also discussed.
