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The neutrinoless double beta decay is a commonly proposed and experimentally pursued theoretical radioactive decay process that would prove a Majorana nature of the neutrino particle. To this day, it has not been found.
The discovery of the neutrinoless double beta decay could shed light on the absolute neutrino masses and on their mass hierarchy. It would mean the first ever signal of the violation of total lepton number conservation. A Majorana nature of neutrinos would confirm that the neutrino's own antiparticle is no different than itself, i.e. it is its own antiparticle.
To search for neutrinoless double beta decay, there are currently a number of experiments underway, with several future experiments for increased sensitivity proposed as well.