With a mass at least six orders of magnitudes smaller than the mass of an electron - but non-zero - neutrinos are a clear misfit in the Standard Model of Particle Physics. On the one hand, its tiny mass makes the neutrino one of the most interesting particles, one that might hold the key to physics beyond the Standard Model. On the other hand this minute mass leads to great challenges in its experimental determination. Three approaches are currently pursued: An indirect neutrino mass determination via cosmological observables, the search for neutrinoless double β-decay, and a direct measurement based on the kinematics of single β- decay. In this paper the latter will be discussed in detail and the status and scientific reach of the current and near-future experiments will be presented.