  We review the recent developments in the field of nuclear double beta decay,
which is presently an important topic in both nuclear and particle physics. The
mechanism of lepton number violation within the neutrinoless double beta decay
is discussed in context of the problem of neutrino mixing and the R-parity
violating supersymmetric extensions of the Standard model. The problem of
reliable determination of the nuclear matrix elements governing both
two-neutrino and neutrinoless modes of the double beta decay is addressed. The
validity of different approximation schemes in the considered nuclear structure
studies is analyzed and the role of the Pauli exclusion principle for a correct
treatment of nuclear matrix elements is emphasized. The constraints on
different lepton number violating parameters like effective electron neutrino
mass, effective right-handed weak interaction parameters, effective Majoron
coupling constant and R-parity violating SUSY parameters are derived from the
best presently available experimental limits on the half life of neutrinoless
mode of this process.
