The recent genealogical history of human populations is a complex mosaic formed by individual migration, large-scale population movements, and other demographic events. Population genomics datasets can provide a window into this recent history, as rare traces of recent shared genetic ancestry are detectable due to long segments of shared genomic material. We make use of genomic data for 2,257 Europeans (in the Population Reference Sample [POPRES] dataset) to conduct one of the first surveys of recent genealogical ancestry over the past 3,000 years at a continental scale. We detected 1.9 million shared long genomic segments, and used the lengths of these to infer the distribution of shared ancestors across time and geography. We find that a pair of modern Europeans living in neighboring populations share around 2-12 genetic common ancestors from the last 1,500 years, and upwards of 100 genetic ancestors from the previous 1,000 years. These numbers drop off exponentially with geographic distance, but since these genetic ancestors are a tiny fraction of common genealogical ancestors, individuals from opposite ends of Europe are still expected to share millions of common genealogical ancestors over the last 1,000 years. There is also substantial regional variation in the number of shared genetic ancestors. For example, there are especially high numbers of common ancestors shared between many eastern populations that date roughly to the migration period (which includes the Slavic and Hunnic expansions into that region). Some of the lowest levels of common ancestry are seen in the Italian and Iberian peninsulas, which may indicate different effects of historical population expansions in these areas and/or more stably structured populations. Population genomic datasets have considerable power to uncover recent demographic history, and will allow a much fuller picture of the close genealogical kinship of individuals across the world.