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1. Positive relationships between the local abundance and the range size of the species in a taxonomic assemblage are very general. 2. Explanations for these relationships typically focus on two mechanisms, based on differences in the niche breadths of species, or metapopulation dynamics. Others have, however, also been suggested. 3. Here we identify and clarify all the principal mechanisms proposed to explain positive interspecific abundance-range size relationships. We critically assess the assumptions and predictions that they make, and the evidence in support of them. 4. A number of predictions are common to all of the biological (as opposed to artefactual) mechanisms, but the combination of predictions and assumptions made by each is unique, suggesting that, in principle, conclusive tests of all of the mechanisms are possible. 5. On present evidence, no single mechanisms has unequivocal support …