Broad-scale variation in taxonomic richness is strongly correlated with climate. Many mechanisms have been hypothesized to explain these patterns; however, testable predictions that would distinguish among them have rarely been derived. Here, we examine several prominent hypotheses for climate–richness relationships, deriving and testing predictions based on their hypothesized mechanisms. The ‘energy–richness hypothesis’ (also called the ‘more individuals hypothesis’) postulates that more productive areas have more individuals and therefore more species. More productive areas do often have more species, but extant data are not consistent with the expected causal relationship from energy to numbers of individuals to numbers of species. We reject the energy–richness hypothesis in its standard form and consider some proposed modifications. The ‘physiological tolerance hypothesis’ postulates that richness varies according to the tolerances of individual species for different sets of climatic conditions. This hypothesis predicts that more combinations of physiological parameters can survive under warm and wet than cold or dry conditions. Data are qualitatively consistent with this prediction, but are inconsistent with the prediction that species should fill climatically suitable areas. Finally, the ‘speciation rate hypothesis’ postulates that speciation rates should vary with climate, due either to faster evolutionary rates or stronger biotic interactions increasing the opportunity for evolutionary diversification in some regions. The biotic interactions mechanism also has the potential to amplify shallower, underlying gradients in richness. Tests of speciation rate hypotheses are few (to date), and their results are mixed.