An interspecific relationship between animal abundance and body size is one of the most frequently reported patterns in ecology. However, there has been little progress towards an understanding of the mechanisms causing the relationship, focus having dwelt on debate over its form. Only one mechanism has been given serious consideration as an explanation for abundance-body size relationships. This is that abundances are constrained by energy availability. However, there are at least five other hypotheses in the literature. Here, we critically assess the extent to which each hypothesis can explain the observed patterns. None of the six hypotheses can be considered adequate as explanations of the abundance-body size relationship. Most importantly, the energetic constraint hypothesis is shown to be logically flawed because, if its predictions hold, then its assumptions must be incorrect (and vice versa). Moreover, it is never possible to prove that abundances are energy-limited. We suggest a new, and more general, model whereby a negative abundance-body size relationship arises as a logical consequence of the observed frequency distribution of biomasses amongst species, and a constraint on minimum viable abundance. However, explanations for what shapes the biomass distribution and how minimum viable abundances are related to body size are both likely to be difficult to obtain.