Abstract: Experiments with solar, atmospheric, reactor and accelerator neutrinos convincingly show that neutrinos are massive and mixed. But why are neutrinos so extremely light? Perhaps their mass has a different origin, and they are Majorana fermions, unlike the charged leptons and quarks that are Dirac fermions? Neutrinoless ββ decay is the most sensitive probe available to test these ideas. I will first review the history and status of the search for this mode of ββ decay. Then I will describe my own work dealing with the evaluation of the nuclear matrix elements that govern the decay rate and the difficulties of such calculations. Finally, I will briefly review the worldwide program of experimental search for the neutrinoless ββ decay and its relation to other searches for neutrino mass.