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One of the most exciting predictions of Einstein's theory of gravitationisthat there may exist 'black holes': putative objects whose gravitational fields are so strong that no physical bodies and signals can break free of their pull and escape. Even though a completely reliable discovery of a black hole has not yet been made, several objects among those scrutinized by astrophysicists will very likely be conformed as black holes. The proof that they do exist, and an analysis of their properties, would have a significance going far beyond astrophysics. Indeed, what is involved is not just the discovery of yet another, even if extremely remarkable, astrophysical object, but a test of the correctness of our understanding the properties of space and time in extremely strong gravitational fields. Theoretical research into the properties of black holes and into the possible corollaries of the hypothesis that they exist, has been carried out with special vigor since the beginning of the 1970s. In addition to those specific features of black holes that are important for the interpretation of their possible astrophysical manifestations, the theory has revealed a nurober of unexpected characteristics of physical interactions involving black holes. By now, a fairly detailed understanding has been achieved of the properties of the black holes, their possible astrophysical manifestations, and the specifics of the various physical processes involved. Furthermore, profound links were found between black-hole theory and such seemingly very distant fields as thermodynamics, information theory, and quantum theory.