## Abstract:

Observing neutrinoless double-beta (0nbb) decay is the most promising

way to detect lepton number violation in the laboratory, and it would imply

that neutrinos are its own antiparticle. The decay half-life naturally

depends on a nuclear matrix element that needs to be calculated

theoretically. A good knowledge of this matrix element is key for the

planning of 0nbb decay experiments, and also to extract information on

the neutrino mass once 0nbb decay is observed.

At the moment, predicted matrix-element values depend on the

many-body method used to calculate them and, in addition, they may

need to be ”quenched”, as the matrix elements of other beta decays that,

however, have a very different momentum-transfer regime. I will discuss

recent efforts towards obtaining reliable nuclear matrix elements, ranging

from improved calculations with standard many-body approaches, to the

first application of "ab initio" many-body methods to 0nbb decay, finalizing

with possible measurements that could be very useful to test calculations

and to constrain the value of the 0nbb matrix elements.