ABSTRACT Detecting adaptive genetic variation in population genomic datasets is important for understanding the genetic architecture underlying complex genetic diseases. Humans and other natural populations have been evolving under complex demographic histories, including divergence of ancestral populations, migration in structured populations, and past population size changes. Adaptive genetic variation and variation subject to complex demographic histories can result in similar observable genomic patterns, and distinguishing the evolutionary forces underlying genetic variation observed in natural population remains challenging. It is thus of importance to unravel the complex demographic histories underlying natural populations, and develop methods that detect adaptive genetic variation while properly accounting for these histories. In addition to contemporary genomic data, researchers have been gathering genetic data from ancient human remains in recent years. Including such datasets into the analyses has the potential to vastly improve our ability to detect population structure and genetic variation adapting to selective pressure. Thus, we will develop several tools for the analysis of contemporary and ancient genomic datasets to unravel the migration histories underlying the population expansion of humans and to detect adaptive genetic variation while accounting for these histories. To this end, we will develop a novel Coalescent Hidden Markov Model method to characterize complex migration histories. Our novel approach will use more efficient representations of local genealogies then previous approaches, which increases the accuracy of the inference and is more robust to noise in the data. Moreover, this framework will allow us to analyze population genomic data from large public databases to identify adaptive genetic variation. The local genealogies will be highly skewed in regions with adaptive genetic variation, as compared to genomic regions evolving under neutrality. The novel framework can be used to compute the posterior distribution of genealogical summaries at different locations in the genome to identify regions with skewed genealogies. In addition, we will implement approaches to detect adaptive genetic variation based on forward-in-time solutions of the dynamics of beneficial genetic variation and linked neutral regions. Based on a previously developed numerical approach, we will develop composite likelihood frameworks of observed genomic sequence variation under this model to detect adaptive genetic variation, while accounting for the underlying complex demographic history. Moreover, we will develop a method that aims at detecting polygenic adaptation from ancient DNA. This approach will be based on explicit likelihood models of the underlying allele frequency dynamics and allow us to detect and quantify directional and, unlike previous approaches, stabilizing selection on complex traits. Lastly, we will collaborate with colleagu...