  Neutrinoless Double Beta Decay ($0\nu\beta\beta$) is presently the only known
experiment to distinguisch between Dirac neutrinos, different from their
antiparticles, and Majorana neutrinos, identical with their antiparticles. In
addition $0\nu\beta\beta$ allows to determine the absolute scale of the
neutrino masses. This is not possible with neutrino oscillations. To determine
the neutrino masses one must assume, that the light Majorana neutrino exchange
is the leading mechanism for $0\nu\beta\beta$ and that the matrix element of
this transition can ba calculated reliably. The experimental $0\nu\beta\beta$
transition amplitude in this mechanism is a product of the light left handed
effective Majorana neutrino mass and of this transition matrix element. The
different methods, Quasi-particle Random Phase Approximation (QRPA), Shell
Model (SM), Projected Hartree-Fock-Bogoliubov (PHFB) and Interacting Boson
Model (IBM2) used in the literature and the reliability of the matrix elements
in these approaches are reviewed. In the second part it is investigated how one
can determine the leading mechanism or mechanisms from the data of the
$0\nu\beta\beta$ decay in different nuclei. Explicite expressions are given for
the transition matrix elements. is shown, that possible interference terms
allow to test CP (Charge and Parity conjugation) violation.
