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Recent developments in the theory of nucleation of vapour deposits on crystalline substrates are reviewed. To facilitate comparison, the theories are formulated in a common dimensionless notation, and an examination of the major underlying assumptions reveals the basic similarity of many of them. The capture and dissociation rates are expressed in terms of the cluster geometry and the pertinent energy parameters, and from these the ‘chemical’ rate equations are set up. The following types of approximate solutions are discussed: (a) long-time asymptotic solutions, from which the conditions for saturation in the cluster concentration may be deduced, (b) a generalization of the type of approximate solution used by Logan (1969), and (c) numerical solutions employing a minimum number of simplifying assumptions. Based on a simple model, agreement between the latter two seems reasonably good. For any given set of fixed parameters (energies, geometrical constants, and arrival rate from the vapour) several temperature ranges may be distinguished. The main division is between ‘initially complete condensation’ at low temperatures and ‘initially incomplete condensation’ at high temperatures. Within each of the latter cases there are further transitions (a) between different values of the ‘critical size’ i*, and (b) from negligible growth to rapid growth of the supercritical clusters. The influence of all of these factors on the final cluster concentration is described.The distributions of the clusters in size and spacing are discussed briefly and qualitatively, as are the types of effects that can be induced by defects or other ‘special sites’ on the substrate.Comparisons are made with some recent experimental studies. In many of these, defects in the substrate seem to play a dominant role, and no detailed comparison with theory seems possible. One notable exception is the nucleation of rare gas crystals on graphite substrates (Venables and Ball 1970), and here, for at least two of the three gases studied, excellent quantitative agreement is obtained.