The observation of neutrinoless double beta decay will have important consequences. First it will signal that lepton number is not conserved and the neutrinos are Majorana particles. Second, it represents our best hope for determining the absolute neutrino mass scale at the level of a few tens of meV. To achieve the last goal, however, certain hurdles have to be overcome involving particle, nuclear and experimental physics. Particle physics is important since it provides the mechanisms for neutrinoless double beta decay. In this review we emphasize the light neutrino mass mechanism. Nuclear physics is important for extracting the useful information from the data. One must accurately evaluate the relevant nuclear matrix elements, a formidable task. To this end, we review the recently developed sophisticated nuclear structure approaches, employing different methods and techniques of calculation. We also examine the question of quenching of the axial vector coupling constant, which may have important consequences on the size of the nuclear matrix elements. From an experimental point of view it is challenging, since the life times are extremely long and one has to fight against formidable backgrounds. One needs large isotopically enriched sources and detectors with good energy resolution and very low background.
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