  With the compelling evidence for massive neutrinos from recent neutrino
oscillation experiments, one of the most fundamental tasks of particle physics
over the next years will be the determination of the absolute mass scale of
neutrinos, which has crucial implications for cosmology, astrophysics and
particle physics. Neutrino oscillation experiments can measure squared mass
differences but not masses. The latter have to be determined in a different
way. The direct mass experiments investigate -- besides time-of-flight
measurements -- the kinematics of weak decays obtaining information on the
neutrino mass without further requirements. Here the tritium beta decay
experiments give the most stringent results. The current tritium beta decay
experiments at Mainz and Troitsk are reaching their sensitivity limit. The
different options for a next generation direct neutrino mass experiment with
sub-eV sensitivity are discussed. The KATRIN experiment, which will investigate
the tritium beta spectrum with a MAC-E-Filter of 1 eV resolution, is being
prepared to reach a sub-eV sensitivity.
