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Since Pauli's hypothesis of their existence in 1930, neutrinos never ceased to bring into play novel ideas and to add new pieces of physics in the whole picture of fundamental interactions. They are only weakly interacting and, at odds with Standard Model's predictions, have a mass less than one millionth of the electron mass, which makes the investigation of their properties very challenging. The issue of the measurement of neutrino's rest mass gained a wider and wider consensus since its discovery through neutrino oscillations in 1998. Various neutrino sources are available for experiments, ranging from nuclear collisions of cosmic rays in the Earth atmosphere and supernova explosions to neutrino beams produced by accelerators and power reactors. These suggest different approaches to the experimental detection and measurement of the absolute value of the neutrino mass. In this paper, we retrace the intriguing story of this endeavor, focusing mainly on direct mass determination methods. The puzzling issue of the nature of massive neutrinos is addressed as well with explicit reference to the phenomenon of double beta-decay as a viable experimental tool to discriminate between Dirac's and Majorana's nature. [ABSTRACT FROM AUTHOR]