We report on a study of the richness dependence of the spatial correlations of clusters of galaxies. We employ the method devised by Bond & Couchman, which combines the theory of the statistics of peaks in Gaussian random fields with the evolution of the cosmological density field by the Zeldovich Approximation, to calculate analytically both the statistical and dynamical contributions to the clustering. We compute the cluster correlation function for a variety of popular cosmological models and compare our results with data from four recent cluster samples. We find no model able to account for all of the observations, although, conversely, the observational data are of insufficient quality to rule out firmly any of the models. The model that fares best is one which has been advocated as accounting for the angular correlation function of APM galaxies: that the same power spectrum gives the best fit to both the galaxy and cluster correlation data may be taken as support for the standard picture, due to Kaiser, in which objects form at the sites of peaks in an initial cosmological density field which obeyed Gaussian statistics. No model is able to reproduce the correlation length of a sample of Abell R >= 2 clusters. This result may just indicate the inadequacy of the correlation length when taken alone as a diagnostic statistic, or that this cluster sample is seriously corrupted by projection effects. We consider, however, alternative explanations, including the possibility that non-Gaussian initial conditions are required and that the identification of peaks in the linear density field as sites of nascent clusters may break down for the highest peaks, corresponding to the richest clusters.