After nearly 80 years since the first guess on its existence, neutrino still
escapes our insight: the mass and the true nature (Majorana or Dirac) of this
particle is still unknown. In the past ten years, neutrino oscillation
experiments have finally provided the incontrovertible evidence that neutrinos
mix and have finite masses. These results represent the strongest demonstration
that the Standard Model of electroweak interactions is incomplete and that new
Physics beyond it must exist. None of these experimental efforts could however
shade light on some of the basic features of neutrinos. Indeed, absolute scale
and ordering of the masses of the three generations as well as charge
conjugation and lepton number conservation properties are still unknown. In
this scenario, a unique role is played by the Neutrinoless Double Beta Decay
searches: these experiments can probe lepton number conservation, investigate
the Dirac/Majorana nature of the neutrinos and their absolute mass scale
(hierarchy problem) with unprecedented sensitivity. Today Neutrinoless Double
Beta Decay faces a new era where large scale experiments with a sensitivity
approaching the so-called degenerate-hierarchy region are nearly ready to start
and where the challenge for the next future is the construction of detectors
characterized by a tonne-scale size and an incredibly low background, to fully
probe the inverted-hierarchy region. A number of new proposed projects took up
this challenge. These are based either on large expansions of the present
experiments or on new ideas to improve the technical performance and/or reduce
the background contributions. n this paper, a review of the most relevant
ongoing experiments is given. The most relevant parameters contributing to the
experimental sensitivity are discussed and a critical comparison of the future
projects is proposed.Comment: 70 pages, 16 figures, 6 tables. arXiv admin note: text overlap with
arXiv:1109.5515, arXiv:hep-ex/0501010, arXiv:0910.2994 by other author
