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Compton scattering is one of the major physical processes that couples matter and radiation. Its importance is often stressed in highly relativistic environments where large energy transfers occur: for example, in the synchrotron self-Compton process that may be responsible for much of the X-radiation from active galactic nuclei (e.g., Fabian et al. 1986). However, the Compton process also has observable consequences in low-energy environments, where small energy transfers occur. The Sunyaev-Zel'dovich effect, which arises from the scattering of electrons in clusters of galaxies on the cosmic microwave background radiation field, is perhaps the most important astrophysical example. The effect provides a cosmological probe, it has been used to measure the properties of gas in clusters of galaxies, and it has been discussed as a means of measuring the motions of clusters of galaxies and hence studying the evolution of structure in the Universe.

The purpose of this review is to provide a comprehensive introduction to the Sunyaev-Zel'dovich effect. I aim to provide both a theoretical treatment that can be followed by non-specialists, and an introduction to the observation of the effect with a critical review of data in the literature. The latter is more difficult today than it would have been five years ago because of the rapid increase in the number of papers on the Sunyaev-Zel'dovich effect, and the improvement in the quality of the results that are being gained.