Prior to 1958, nearly all astronomical observations could be interpreted in terms of the major observed constituents of the Universe, ordinary stars and interstellar gas. However, the realization that the energy released within strong radio sources can exceed 1060 ergs (Burbidge 1959; Maltby, Matthews & Moffet 1963), plus the evidence for their origin in a violent explosion in a galactic nucleus (Lynds & Sandage 1963; Burbidge, Burbidge & Sandage 1963), led many astrophysicists to the belief that such events are produced by a new class of objects, possibly involving strong gravitational fields (Robinson, Schild & Schucking 1965). This belief has been strengthened by the discovery and interpretation of the quasistellar objects (Matthews & Sandage 1963; Hazard, Mackey & Shimmins 1963; Schmidt 1963; Burbidge & Burbidge 1967) and by recent observations of the activity within the nuclei of Seyfert galaxies (Burbidge & Burbidge 1965, Walker 1968, Rubin & Ford 1968) , which appear in many respects similar to quasistellar objects. Hoyle & Fowler (1963a,b) were the first to propose a specific model for the object responsible, visualizing it as a star of very great mass, M.?:,105M0• Subsequently, there have been many investigations of the possibility of re leasing large amounts of nuclear or gravitational energy from such bodies. However, it is also of considerable interest to study the properties of such large masses in a more general way. It is the purpose of this paper to review the theory of objects, with a view toward describing their (directly or indirectly) observable characteristics. This study will be restricted to coherent objects which spend at least some time in an equilibrium state. Thus we shall not consider masses which emerge from a state of extreme density, such as investigated by Novikov (1964) and Ne'eman (1965), or dense clusters of stars (Zel'dovich & Podurets 1965, Fackerell 1968, Ipser & Thorne 1968) which have been proposed (Gold, Axford & Ray 1965; Colgate 1967; Hoyle & Fowler 1967) to play various roles in quasistellar objects. Finally, the analyses will be limited to masses M,2:,10(M0, which includes the range of supermassive stars, but excludes what are usually referred to as massive stars. We shall concentrate on the best-understood aspects of such bodies, with only some references to unsupported speculations. Most of what follows will therefore be concerned with the equilibrium phases of evolution, for which the most detailed calculations exist. In the presentation of the results, an