  Neutrinoless double beta decay is a very important process both from the
particle and nuclear physics point of view. From the elementary particle point
of view it pops up in almost every model. giving rise, among others, to the
following mechanisms: a) The traditional contributions like the light neutrino
mass mechanism . b) The exotic R-parity violating supersymmetric (SUSY)
contributions. Similar considerations apply to its sister lepton and muon
number violating muon to positron conversion. The observation of neutrinoless
double beta decay will signal that the neutrinos are massive Majorana
particles. For nuclear physics it is challenging, because: 1) The nuclei, which
can undergo double beta decay, have complicated structure. 2) The energetically
allowed transitions are suppressed. 3) One must cope with the short distance
behavior of the transition operators. Thus novel effects (decay of pions in
flight between nucleons etc) have to be considered. 4) One has to take into
account possible momentum dependent terms of the nucleon current. For light
neutrinos such modifications of the nucleon current reduce the nuclear matrix
elements by about 25 per cent. In the case of heavy neutrino the effect is much
larger and model dependent. With the above effects the nuclear matrix elements
for all interesting nuclei A = 76, 82, 96, 100, 116, 128, 130, 136 and 150 are
discussed. Some have been obtained in the shell model but most in QRPA. Thus we
have extracted new limits on the various lepton violating parameters. We find
an average neutrino mass of about 0.5eV and, for reasonable choices of the
parameters of SUSY models, we get a stringent limit on the R-parity violating
parameter less than 0.00068.
