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 (the POPRES dataset) to conduct one of the first surveys of recent genealogical ancestry over the past three thousand years at a continental scale. We detected 1.9 million shared 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 10-50 genetic common ancestors from the last 1500 years, and upwards of 500 genetic ancestors from the previous 1000 years. These numbers drop off exponentially with geographic distance, but since genetic ancestry is rare, individuals from opposite ends of Europe are still expected to share millions of common genealogical ancestors over the last 1000 years. There is substantial regional variation in the number of shared genetic ancestors: especially high numbers of common ancestors between many eastern populations likely date to the Slavic and/or Hunnic expansions, while much lower levels of common ancestry in the Italian and Iberian peninsulas may indicate weaker demographic effects of Germanic expansions into these areas and/or more stably structured populations. Recent shared ancestry in modern Europeans is ubiquitous, and clearly shows the impact of both small-scale migration and large historical events. 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.