  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.
