Before we can be sure we have a dark-matter problem we have to first be certain that no known particle can account for the missing matter. The last possibility has long been the neutrino, which, while massless in the Standard Model of particle physics, is the second most numerous particle in the Universe (after the photon) and thus (if massive) a potential source of substantial unaccounted for mass. Recent neutrino oscillation measurements have, in fact, confirmed that the Standard Model is incomplete and that neutrinos have mass. However, recent measurements have confirmed that the resulting mass is insufficient for neutrinos to make up the bulk of the dark matter. In fact, observations of the matter distribution in the Universe are now competing with laboratory measurements in their sensitivity to the absolute masses of neutrinos. The article discusses all these measurements and gives some guesses about where we may get in our measurements of neutrino masses in the future.