Population expansions have a major evolutionary impact on the genetic diversity of species. When a population expands into a new territory, it often encounters other populations of the same or closely related species. Interbreeding between the migrating population and the local one occurs can generate specific molecular patterns in both populations. These patterns can potentially be used to infer the migratory and demographic dynamics of the expansion. However, the effect of interbreeding during population expansions on the spatiotemporal patterns of molecular diversity in the involved populations has not been studied yet. In the present thesis, I used palaeogenomic data to investigate spatiotemporal patterns of molecular diversity that were created during past expansions of the species Homo sapiens, and I used these patterns to make inferences about these important evolutionary processes. The thesis mainly focused on patterns of genomic introgression. In Chapter 2, we used spatially explicit simulations to investigate the expected pattern of introgression after an expansion with interbreeding. We showed that the levels of introgression from the local population in the expanding one are expected to increase with distance from the source of the expansion, while the opposite cline is expected in the local population. We reported the existence of such empirical introgression patterns in various species that underwent an expansion with interbreeding, in accordance with our theoretical expectation. In Chapter 3, we used palaeogenomic data to investigate the spatiotemporal pattern of Neanderthal introgression in Eurasian anatomically modern human populations at different periods of time. In accordance with the theoretical expectations from Chapter 2, we found that during the expansion of modern humans in Eurasia, a spatial cline of Neanderthal introgression developed, increasing with distance from the Near East, and it persists until the present. In Chapter 4, we inferred the population dynamics of two human migrations that created the observed spatiotemporal pattern of Neanderthal introgression in Europe, by using a novel spatially explicit simulation framework of three populations. We showed that the observed cline of Neanderthal introgression is indicative of the direction from which the first humans arrived in Europe and that the reduction of Neanderthal introgression levels during the Neolithic transition can be explained by drift, and demographic and migratory processes. Lastly, in Chapter 5, we investigated in more depth the Neolithic transition in Europe, by following two different approaches and using two palaeogenomic datasets. Our results are compatible with a model of “demic diffusion with delayed admixture”, meaning that early farmer populations migrated into Europe from Anatolia and that the assimilation of local hunter-gatherers increased with time. Altogether, my thesis shows that population expansions with interbreeding create clear spatiotemporal patterns of local introgression in the migrating population, which can provide information both on the direction of past expansions and on the demography and interactions of the involved populations. Although focused on specific events of human evolution, our approach and results offer a framework for using the introgression levels in genomic data to study other instances of spatial processes between populations, in humans or other species.