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Humans like to tell stories. Amongst the most captivating is the story of the global spread of modern humans from their original homeland in Africa. Traditionally this has been the preserve of anthropologists, but geneticists are starting to make an important contribution. However, genetic evidence is typically analyzed in the context of anthropological preconceptions. For genetics to provide an accurate and detailed history without reference to anthropology, methods are required that translate DNA sequence data into histories. We introduce a statistical method that has three virtues. First, it is based on a copying model that incorporates the block-by-block inheritance of DNA from one generation to the next. This allows it to capture the rich information provided by patterns of DNA sharing across the whole genome. Second, its parameter space includes an enormous number of possible colonization scenarios, meaning that inferences are correspondingly rich in detail. Third, the inferred colonization scenario is determined algorithmically. We have applied this method to data from 53 human populations and find that while the current consensus is broadly supported, some populations have surprising histories. This scenario can be viewed as a movie, making it transparent where statistical analysis ends and where interpretation begins.