This paper reviews recent experiments on tritium β-spectroscopy searching for the absolute value of the electron neutrino mass m(νe). By the use of dedicated electrostatic filters with high acceptance and resolution, the uncertainty on the observable m2(νe) has been pushed down to about 3 eV2. The new upper limit of the mass is m(νe) < 2 eV at 95% C.L. In view of erroneous and unphysical mass results obtained by some earlier experiments in β decay, particular attention is paid to systematic effects. The mass limit is discussed in the context of current neutrino research in particle- and astrophysics. A preview is given of the next generation of β-spectroscopy experiments currently under development and construction; they aim at lowering the m2(νe) uncertainty by another factor of 100, reaching a sensitivity limit m(νe) < 0.2 eV.