  We study the impact of light gauge bosons on neutrino physics. We show that
they can explain the NuTeV anomaly and also escape the constraints from
neutrino experiments if they are very weakly coupled and have a mass of a few
GeV. Lighter gauge bosons with stronger couplings could explain both the NuTeV
anomaly and the positive anomalous magnetic moment of the muon. However, in the
simple model we consider in this paper (say a purely vectorial extra U(1)
current), they appear to be in conflict with the precise measurements of \nu-e
elastic scattering cross sections. The surprising agreement that we obtain
between our naive model and the NuTeV anomaly for a Z' mass of a few GeV may be
a coincidence. However, we think it is interesting enough to deserve attention
and perhaps a more careful analysis, especially since a new light gauge boson
is a very important ingredient for the Light Dark Matter scenario.
