The possibility that atomic nuclei can coexist in different shapes adds to the richness of structural effects found in the nuclear many-body problem. Shape coexistence means that the nucleus can exist in different minima of an energy potential. Ultimately, in regions where these minima compete at the lowest energy, the switch of the ground-state deformation from one to another within a chain of isotopes leads to nuclear shape-phase transitions. New theoretical approaches exist, which are tested by new experiments with increasing sensitivity and today often reach into regions of the nuclear chart which have just become accessible through new facilities and instrumentation. Selected recent work will be discussed, and focus will be given to implications for isotopes which are used in the search for neutrino-less double-beta decay, which have recently be studied using photon-scattering techniques.