Constraints on the Active and Sterile Neutrino Masses from Beta-Ray Spectra: Past, Present and Future

Although neutrinos are probably the most abundant particles of the universe their mass is not yet known. Oscillation experiments have proven that at least one of the neutrino mass states has m_{i}>0.05 eV while various interpretations of cosmological observations yielded an upper limit for the sum of neutrino masses \sum m_{i}<(0.14-1.7) eV. The searches for the yet unobserved 0\nu\beta\beta decay result in an effective neutrino mass m_{\beta\beta}<(0.2-0.7) eV. The analyses of measured tritium \beta-spectra provide an upper limit for the effective electron neutrino mass m(\nu_e)<2 eV. In this review, we summarize the experience of two generations of \beta-ray spectroscopists who improved the upper limit of m(\nu_e) by three orders of magnitude. We describe important steps in the development of radioactive sources and electron spectrometers, and recapitulate the lessons from now-disproved claims for the neutrino mass of 30 eV and the 17 keV neutrino with an admixture larger than 0.03 %. We also pay attention to new experimental approaches and searches for hypothetical sterile neutrinos.