The article presents a suite of methods for studying population mixture, implemented in the software package ADMIXTOOLS. These methods formally test for the occurrence of mixture and allow inference of mixture proportions and dates. The authors describe a new SNP array designed for population genetic analyses and apply their methods to 934 individuals from 53 diverse populations. A key finding is a clear signal of admixture into northern Europe, involving populations related to present-day Basques, Sardinians, and northeast Asians and the Americas. This suggests a history of admixture between Neolithic migrants and indigenous Mesolithic populations in Europe, consistent with recent archaeological and genetic evidence. The methods include the three-population test, $D$-statistics, $F_{a}$-ratio estimation, admixture graph fitting, and rolloff, each designed to address specific aspects of population mixture. The article also discusses the robustness of these methods to ascertainment processes and provides theoretical and empirical demonstrations of their effectiveness.The article presents a suite of methods for studying population mixture, implemented in the software package ADMIXTOOLS. These methods formally test for the occurrence of mixture and allow inference of mixture proportions and dates. The authors describe a new SNP array designed for population genetic analyses and apply their methods to 934 individuals from 53 diverse populations. A key finding is a clear signal of admixture into northern Europe, involving populations related to present-day Basques, Sardinians, and northeast Asians and the Americas. This suggests a history of admixture between Neolithic migrants and indigenous Mesolithic populations in Europe, consistent with recent archaeological and genetic evidence. The methods include the three-population test, $D$-statistics, $F_{a}$-ratio estimation, admixture graph fitting, and rolloff, each designed to address specific aspects of population mixture. The article also discusses the robustness of these methods to ascertainment processes and provides theoretical and empirical demonstrations of their effectiveness.