The origins of the Bronze Age Minoan and Mycenaean cultures have puzzled archaeologists for more than a century. We have assembled genome-wide data from 19 ancient individuals, including Minoans from Crete, Mycenaeans from mainland Greece, and their eastern neighbours from southwestern Anatolia. Here we show that Minoans and Mycenaeans were genetically similar, having at least three-quarters of their ancestry from the first Neolithic farmers of western Anatolia and the Aegean, and most of the remainder from ancient populations related to those of the Caucasus and Iran. However, the Mycenaeans differed from Minoans in deriving additional ancestry from an ultimate source related to the hunter-gatherers of eastern Europe and Siberia, introduced via a proximal source related to the inhabitants of either the Eurasian steppe or Armenia. Modern Greeks resemble the Mycenaeans, but with some additional dilution of the Early Neolithic ancestry. Our results support the idea of continuity but not isolation in the history of populations of the Aegean, before and after the time of its earliest civilizations.