Identifications with a relatively complete sample of high-latitude X-ray sources, including four new identifications with Abell clusters, are used to establish the luminosity function for extragalactic sources, and to determine the maximum rates of evolution for different populations of source consistent with the observed integrated X-ray background. A significant limit is obtained on any evolution that operates on all sources. Without evolution, quasars, and radio and normal galaxies make a negligible contribution to the background. Seyferts may contribute about 30 per cent, but with an uncertainty of the same order. Rich clusters contribute about 10 per cent, but make up over 80 per cent of the identifications. Both inverse Compton and thermal Bremsstrahlung models of cluster sources are investigated, and severe restrictions can be placed on models of Silk & Tarter, in which a strong correlation of X-ray luminosity with richness is predicted. We are sceptical of reported correlations of the X-ray luminosity of clusters with radio luminosity or morphological type.