The present state of the problem of the double neutrinoless beta decay, 2$\beta$(0ν), and related questions is reviewed. Various mechanisms for $\Delta L=2$ transitions in models with spontaneous violation of lepton number $(L)$ conservation are discussed. From the standpoint of unified gauge theories, a Majorana neutrino mass would be the most probable reason for processes with $\Delta L=2$ (if they occur). The 2$\beta$(0ν) decay is compared with other phenomena which are sensitive to the neutrino mass. It follows from this comparison that the 2$\beta$(0ν) decay is essentially the only process which could be detected at the present level of experimental capabilities and which might tell us about the nature of the neutrino mass, if this mass is some tens of electron volts.