In this paper, we study the length distribution of tracts of identity by state (IBS), which are the gaps between pairwise differences in an alignment of two DNA sequences. These tract lengths contain information about the amount of genetic diversity that existed at various times in the history of a species and can therefore be used to estimate past population sizes. IBS tracts shared between DNA sequences from different populations also contain information about population divergence and past gene flow. By looking at IBS tracts shared within Africans and Europeans, as well as between the two groups, we infer that the two groups diverged in a complex way over more than 40,000 years, exchanging DNA as recently as 12,000 years ago. Besides having anthropological importance, the history we infer predicts the distribution of pairwise differences between humans extremely accurately at a fine-scale level, which may aid future scans for natural selection in the genome. Despite our current focus on human data, the method is general enough to use on other organisms and has the potential to shed light on the demography of many more species.