  Compelling evidences in favor of neutrino masses and mixing obtained in the
last years in Super-Kamiokande, SNO, KamLAND and other neutrino experiments
made the physics of massive and mixed neutrinos a frontier field of research in
particle physics and astrophysics. There are many open problems in this new
field. In this review we consider the problem of the absolute values of
neutrino masses, which apparently is the most difficult one from the
experimental point of view. We discuss the present limits and the future
prospects of beta-decay neutrino mass measurements and neutrinoless double-beta
decay. We consider the important problem of the calculation of nuclear matrix
elements of neutrinoless double-beta decay and discuss the possibility to check
the results of different model calculations of the nuclear matrix elements
through their comparison with the experimental data. We discuss the upper bound
of the total mass of neutrinos that was obtained recently from the data of the
2dF Galaxy Redshift Survey and other cosmological data and we discuss future
prospects of the cosmological measurements of the total mass of neutrinos. We
discuss also the possibility to obtain information on neutrino masses from the
observation of the ultra high-energy cosmic rays (beyond the GZK cutoff).
Finally, we review the main aspects of the physics of core-collapse supernovae,
the limits on the absolute values of neutrino masses from the observation of
SN1987A neutrinos and the future prospects of supernova neutrino detection.
