The Standard Model predicts that neutrinoless double beta decay does not occur, because it involves lepton number violation, but many beyond the Standard Model theories, including those in which neutrinos have Majorana mass, predict that neutrinoless double beta decay occurs but it merely very rare, which the frequency being a function of the Majorana mass of the neutrino.
The NEMO-3 experiment using a sample of the isotype Molybdenum-100 sees no instances of neutrinoless double beta decay and sets a minimum half-life on the process of 1.1*1024 years at a 90% confidence interval.
This does not exceed the limits set at GERDA of 2.1*1025 years, in the summer of 2014, which is about 20 times longer, and the most strict limits to date. The next phase of the GERDA experiment will increase its sensitivity by a factor of ten.
The universe is roughly 1.4*109 years old, so the current limit from GERDA means that no more than one in 1.5*1016 of hadrons that could experience neutrinoless double beta decay since the formation of the universe have actually done so.
But, each neutrinoless double decay experiment's null results makes the null result more robust (i.e. not subject to the peculiar methodology issues in any particular experiment) and incrementally increases the combined limit since multiple experiments which all fail to detect neutrinoless double beta decay can all be combined to show the total time that observations have been conducted without seeing the phenomena.
Neutrinoless double beta decay also constrains the maximum Majorana mass of the neutrino. The limit of less than 0.62 eV set by the NEMO-3 experiment, which is not as strict as those set by cosmology measurements or other neutrinoless beta decay experiments like GERDA, but still forecloses a great deal of parameter space for beyond the Standard Model theories that permit limited lepton number violation.