  Comprehensive description of the phenomenology of the $\beta\beta$ decay is
given, with emphasis on the nuclear physics aspects. After a brief review of
the neutrino oscillation results and of motivation to test the lepton number
conservation, the mechanism of the $0\nu\beta\beta$ is discussed. Its relation
to the lepton flavor violation involving charged leptons and its use as a
diagnostic tool of the $0\nu\beta\beta$ mechanism is described. Next the basic
nuclear physics of both $\beta\beta$-decay modes is presented, and the decay
rate formulae derived. The nuclear physics methods used, the nuclear shell
model and the quasiparticle random phase approximation, are described next, and
the choice of input parameters is discussed in the following section. Finally,
the numerical values of the nuclear matrix elements, and their uncertainty, are
presented. In the appendix the relation of the search for the neutrino magnetic
moment to the Dirac versus Majorana nature of neutrinos is described.
