  The case for small neutrino mass differences from atmospheric and solar
neutrino oscillation experiments has become compelling, but leaves the overall
neutrino mass scale m_nu undetermined. The most restrictive limit of m_nu < 0.8
eV arises from the 2dF galaxy redshift survey in conjunction with the standard
theory of cosmological structure formation. A relation between the hot dark
matter fraction and m_nu depends on the cosmic number density n_nu of
neutrinos. If solar neutrino oscillations indeed correspond to the favored
large mixing angle MSW solution, then big-bang nucleosynthesis gives us a
restrictive limit on all neutrino chemical potentials, removing the previous
uncertainty of n_nu. Therefore, a possible future measurement of m_nu will
directly establish the cosmic neutrino mass fraction Omega_nu. Cosmological
neutrinos with sub-eV masses can play an interesting role for producing the
highest-energy cosmic rays (Z-burst scenario). Sub-eV masses also relate
naturally to leptogenesis scenarios of the cosmic baryon asymmetry.
Unfortunately, the time-of-flight dispersion of a galactic or local-group
supernova neutrino burst is not sensitive in the sub-eV range.
