  Neutrinoless double beta decay is a very important process both from the
particle and nuclear physics point of view. Its observation will severely
constrain the existing models and signal that the neutrinos are massive
Majorana particles. From the elementary particle point of view it pops up in
almost every model. In addition to the traditional mechanisms, like the
neutrino mass, the admixture of right handed currents etc, it may occur due to
the R-parity violating supersymmetric (SUSY) interactions. From the nuclear
physics point of view it is challenging, because: 1) The relevant nuclei have
complicated nuclear structure. 2) The energetically allowed transitions are
exhaust a small part of all the strength. 3) One must cope with the short
distance behavior of the transition operators, especially when the intermediate
particles are heavy (eg in SUSY models). Thus novel effects, like the double
beta decay of pions in flight between nucleons, have to be considered. 4) The
intermediate momenta involved are about 100 MeV. Thus one has to take into
account possible momentum dependent terms in the nucleon current. We find that,
for the mass mechanism, such modifications of the nucleon current for light
neutrinos reduce the nuclear matrix elements by about 25 per cent, almost
regardless of the nuclear model. In the case of heavy neutrinos the effect is
much larger and model dependent.
  Taking the above effects into account, the available nuclear matrix elements
for the experimentally interesting nuclei A = 76, 82, 96, 100, 116, 128, 130,
136 and 150 and the experimental limits on the life times we have extracted new
stringent limits on the average neutrino mass and on the R-parity violating
coupling for various SUSY models.
