Towards absolute neutrino masses

Various ways of determining the absolute neutrino masses are briefly reviewed and their sensitivities compared. The apparent tension between the announced but unconfirmed observation of the 0νββ decay and the neutrino mass upper limit based on observational cosmology is used as an example of what could happen eventually. The possibility of a 'nonstandard' mechanism of the 0νββ decay is stressed and the ways of deciding which of the possible mechanisms is actually operational are described. The importance of the 0νββ nuclear matrix elements is discussed and their uncertainty estimated