Empowered by modern genotyping and large samples, population structure can be accurately described and quantified even when it only explains a fraction of a percent of total genetic variance. This is especially relevant and interesting for humans, where fine-scale population structure can both confound disease-mapping studies and reveal the history of migration and divergence that shaped our species' diversity. Here we review notable recent advances in the detection, use, and understanding of population structure. Our work addresses multiple areas where substantial progress is being made: improved statistics and models for better capturing differentiation, admixture, and the spatial distribution of variation; computational speed-ups that allow methods to scale to modern data; and advances in haplotypic modeling that have wide ranging consequences for the analysis of population structure. We conclude by outlining four important open challenges: The limitations of discrete population models, uncertainty in individual origins, the incorporation of both fine-scale structure and ancient DNA in parametric models, and the development of efficient computational tools, particularly for haplotype-based methods.
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