A major goal of evolutionary biology is to understand why clades differ dramatically in species richness. A key to this challenge is to uncover the correlates of variation in diversification rate (speciation - extinction) among clades. Here, we explore the relationship between diversification rates and the climatic niches of species and clades among 92 families of terrestrial mammals. We use a time-calibrated molecular phylogeny of mammals and climatic data from 3335 species. We show that considerable variation in net diversification rates among mammal families is explained by niche divergence (59%) and rates of niche change (51%). Diversification rates in turn explain most variation in species richness among families (79%). Contrary to expectations, patterns of diversification are not explained by differences in geographic range areas of clades, nor by their climatic niche position (i.e. whether they are primarily tropical or temperate). Overall, these results suggest that speciation through climatic niche divergence may help drive large-scale patterns of diversification and richness. Our results help explain diversification patterns in a major clade of vertebrates, and suggest that similar underlying principles may explain the diversification of many terrestrial clades.