  A review is given of hypothetical faster-than-light tachyons and the
development of the author's $3+3$ model of the neutrino mass states, which
includes one tachyonic mass state doublet. Published empirical evidence for the
model is summarized, including an interpretation of the mysterious Mont Blanc
neutrino burst from SN 1987A as being due to tachyonic neutrinos having
$m^2=-0.38 eV^2.$ This possibility requires an 8 MeV antineutrino line from SN
1987A, which a new dark matter model has been found to support. Furthermore,
this dark matter model is supported by several data sets: $\gamma-$rays from
the galactic center, and the Kamiokande-II neutrino data on the day of SN
1987A. The KATRIN experiment should serve as the unambiguous test of the $3+3$
model and its tachyonic mass state.
