Pair-instability evolution and explosions in massive stars

Very massive stars are radiation pressure dominated. Before running out of viable nuclear fuel, they can reach a thermodynamic state where electron-positron pair-production robs them of radiation support, triggering their collapse. Thermonuclear explosion(s) in the core ensue. These have long been predicted to result in either repeated episodic mass loss (pulsational pair instability), which reduces the mass available to eventually form a black hole, or, if sufficient energy is generated, the complete unbinding of all stellar material in one single explosive episode (pair instability supernova), which leaves behind no black hole. Despite theoretical agreement among modelers, the wide variety of predicted signatures and the rarity of very high-mass stellar progenitors have so far resulted in a lack of observational confirmation. Nevertheless, because of the impact of pair instability evolution on black hole masses relevant to gravitational-wave astronomy, as well as the present and upcoming expanded capabilities of time-domain astronomy and high redshift spectroscopy, interest in these explosion remains high. We review the current understanding of pair instability evolution, with particular emphasis on the known uncertainties. We also summarize the existing claimed electromagnetic counterparts and discuss prospects for future direct and indirect searches.