Modern humans arrived in Europe similar to 45,000 years ago, but little is known about their genetic composition before the start of farming similar to 8,500 years ago. Here we analyse genome-wide data from 51 Eurasians from similar to 45,000-7,000 years ago. Over this time, the proportion of Neanderthal DNA decreased from 3-6% to around 2%, consistent with natural selection against Neanderthal variants in modern humans. Whereas there is no evidence of the earliest modern humans in Europe contributing to the genetic composition of present-day Europeans, all individuals between similar to 37,000 and similar to 14,000 years ago descended from a single founder population which forms part of the ancestry of present-day Europeans. An similar to 35,000-year-old individual from northwest Europe represents an early branch of this founder population which was then displaced across a broad region, before reappearing in southwest Europe at the height of the last Ice Age similar to 19,000 years ago. During the major warming period after similar to 14,000 years ago, a genetic component related to present-day Near Easterners became widespread in Europe. These results document how population turnover and migration have been recurring themes of European prehistory.