Aim: To investigate the historical distribution of the Cerrado across Quaternary climatic fluctuations and to generate historical stability maps to test: (1) whether the 'historical climate' stability hypothesis explains squamate reptile richness in the Cerrado; and (2) the hypothesis of Pleistocene connections between savannas located north and south of Amazonia. Location: The Cerrado, a savanna biome and a global biodiversity hotspot distributed mainly in central Brazil. Methods: We generated occurrence datasets from 1000 presence points randomly selected from the entire distribution of the Cerrado, as determined by two spatial definitions. We modelled the potential Cerrado distribution by implementing a maximum-entropy machine-learning algorithm across four time projections: current, mid-Holocene (6 ka), Last Glacial Maximum (LGM, 21 ka) and Last Interglacial (LIG, 120 ka). We generated historical stability maps (refugiai areas) by overlapping presence/absence projections of all scenarios, and checked consistencies with qualitative comparisons with available fossil pollen records. We built a spatially explicit simultaneous autoregressive model to explore the relationship between current climate, climatic stability, and squamate species richness. Results: Models predicted the LGM and LIG as the periods of narrowest and widest Cerrado distributions, respectively, and were largely corroborated by palynological evidence. We found evidence for two savanna corridors (eastern coastal during the LIG, and Andean during the LGM) and predicted a large refugiai area in the north-eastern Cerrado (Serra Geral de Goiás refugium). Variables related to climatic stability predicted squamate richness better than present climatic variables did. Main conclusions: Our results indicate that Bolivian savannas should be included within the Cerrado range and that the Cerrado's biogeographical counterparts are not Chaco and Caatinga but rather the disjunct savannas of the Guyana shield plateaus. Climatic stability is a good predictor of Cerrado squamate richness, and our stability maps could be used in future studies to test diversity patterns and genetic signatures of different taxonomic groups and as a higher-order landscape biodiversity surrogate for conservation planning.