Observation of neutrino oscillations implies that neutrinos have mass, but one does not know whether they are described by Dirac or Majorana mass terms. The discovery of a Majorana mass would provide a test of the global symmetries of the standard model, and would have implications for the origin of the number of baryons in the Universe. To achieve this one can search for neutrinoless double-beta decay, a rare nuclear decay where two new electrons are created with no antimatter. The next generation of experiments will explore large Majorana neutrino mass values, also testing alternative theoretical models. In this review, devoted to neutrinoless double-beta decay, the particle and nuclear physics aspects involved in predicting the decay rate and the experimental search methods are discussed.