The increasing abundance of DNA sequences obtained from fossils calls for new population genetics theory that takes account of both the temporal and spatial separation of samples. Here we exploit the relationship between Wright's FST and average coalescence times to develop an analytic theory describing how FST depends on both the distance and time separating pairs of sampled genomes. We apply this theory to several simple models of population history. If there is a time series of samples, partial population replacement creates a discontinuity in pairwise FST values. The magnitude of the discontinuity depends on the extent of replacement. In stepping-stone models, pairwise FST values between archaic and present-day samples reflect both the spatial and temporal separation. At long distances, an isolation by distance pattern dominates. At short distances, the time separation dominates. Analytic predictions fit patterns generated by simulations. We illustrate our results with applications to archaic samples from European human populations.We compare present-day samples with a pair of archaic samples taken before and after a replacement event.. CC-BY 4.0 International license It is made available under a was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.The copyright holder for this preprint (which . http://dx.doi.org/10.1101/362053 doi: bioRxiv preprint first posted online Jul. 4, 2018; 3 Genomic sequences obtained from fossils provide new information about the history of present-day species. Already, thousands of partial or complete genomic sequences have been obtained from modern humans and their extinct relatives, and DNA sequences from fossils of numerous other species have been obtained as well.Population genetics theory of ancient DNA (aDNA) has focused primarily on the time dimension. Several methods have been developed to test for natural selection and estimate selection coefficients in a time series of samples (Bollback, et al. 2008; Malaspinas, et al. 2012; Terhorst, et al. 2015;Schraiber, et al. 2016). Much less effort has gone into incorporating the spatial dimension. The usual approach to analyzing aDNA is to use methods such as principal components analysis (PCA) and f-statistics that were developed for contemporaneous populations and ignore the ages of the fossils from which sequences are obtained. (Slatkin 2016) There are three papers that have considered the spatial and temporal components of aDNA together. Skoglund et al. (2014) developed the coalescent theory of samples of different age and showed that PCA analysis can reveal the time separation of spatially distributed samples. Duforet-Frebourg and Slatkin (2016) extended the classic Kimura-Weiss (1964) analysis of isolation by distance in a stepping-stone model to predict the decrease in identity by descent with increasing spatial and temporal separation of samples. Silva et al. (2017) carried out an extensive simulation study that showed the importance of consi...