Abstract Sheep was one of the first domesticated animals in Neolithic West Eurasia. The zooarchaeological record suggests that domestication first took place in Southwest Asia, although much remains unresolved about the precise location(s) and timing(s) of earliest domestication, or the post-domestication history of sheep. Here, we present 24 new partial sheep paleogenomes, including a 13,000-year-old Epipaleolithic Central Anatolian wild sheep, as well as 14 domestic sheep from Neolithic Anatolia, two from Neolithic Iran, two from Neolithic Iberia, three from Neolithic France, and one each from Late Neolithic/Bronze Age Baltic and South Russia, in addition to five present-day Central Anatolian Mouflons and two present-day Cyprian Mouflons. We find that Neolithic European, as well as domestic sheep breeds, are genetically closer to the Anatolian Epipaleolithic sheep and the present-day Anatolian and Cyprian Mouflon than to the Iranian Mouflon. This supports a Central Anatolian source for domestication, presenting strong evidence for a domestication event in SW Asia outside the Fertile Crescent, although we cannot rule out multiple domestication events also within the Neolithic Fertile Crescent. We further find evidence for multiple admixture and replacement events, including one that parallels the Pontic Steppe-related ancestry expansion in Europe, as well as a post-Bronze Age event that appears to have further spread Asia-related alleles across global sheep breeds. Our findings mark the dynamism of past domestic sheep populations in their potential for dispersal and admixture, sometimes being paralleled by their shepherds and in other cases not.