  The turn of the 21st century witnessed a sudden shift in our fundamental
understanding of particle physics. While the minimal Standard Model predicts
that neutrino masses are exactly zero, the discovery of neutrino oscillations
proved the Standard Model wrong. Neutrino oscillation measurements, however, do
not shed light on the scale of neutrino masses, nor the mechanism by which
those are generated. The neutrino mass scale is most directly accessed by
studying the energy spectrum generated by beta decay or electron capture -- a
technique dating back to Enrico Fermi's formulation of radioactive decay. In
this Article, we review the methods and techniques -- both past and present --
aimed at measuring neutrino masses kinematically. We focus on recent
experimental developments that have emerged in the past decade, overview the
spectral refinements that are essential in the treatment of the most sensitive
experiments, and give a simple yet effective protocol for estimating the
sensitivity. Finally, we provide an outlook of what future experiments might be
able to achieve.
