We investigated the spatial distributions of densities within the geographical ranges of species of carabid beetles, birds and plants. Unselected data of hundreds of species from censuses in sampling areas of varying size and location in northwestern Europe were analysed. The geographical distribution of species abundance is as a rule heterogeneous, with the highest abundances occurring near the centres of the species ranges and the lowest at the margins. As this trend was found in a great number of unselected species belonging to quite different biological taxa, we feel that this trend may be considered as a general biogeographical rule. We obtained this result mainly from indirect measurements, meaning that we deduce the geographical trend within species ranges from correlations of local frequencies of occurrence with the distance of the sampling area to the range centre for each species. The picture may be com- plicated by such biological properties as the species' feeding habits that also greatly affect their density, but within each of the groups distinguished we found that the same general rule applies. The interpretation of such peaked distributions in terms of two-dimensional ecological tolerance or optimum response surfaces in geographic space is discussed in a second paper.