The theoretical foundation of the Sunyaev-Zel'dovich effect was laid in the early 1970s (Sunyaev & Zel'dovich 1970), but is based on earlier work on the interactions of photons and free electrons (Kompaneets 1956; Dreicer 1964; Weymann 1965). Excellent recent reviews of the physics of the Sunyaev-Zel'dovich effect have been given by Bernstein & Dodelson (1990) and Rephaeli (1995b), while discussions of the more general problem of comptonization of a radiation field by passage through an ionized gas have been given by Blumenthal & Gould (1970), Sunyaev & Zel'dovich (1980a), Pozdnyakov et al. (1983), and Nagirner & Poutanen (1994). Comptonization is also an essential ingredient in the discussion of the X-ray and gamma-ray emission of active galactic nuclei (see, for example, Zbyszewska & Zdziarski 1991; Zdziarski et al. 1993; Skibo et al. 1995). The present section relies heavily on this work, and on the material on inverse-Compton scatterings in Rybicki & Lightman (1980), and the papers by Wright (1979) and Taylor & Wright (1989).