Around the turn of the current millennium, the domestication of the four animals that we continue depend on today, sheep, goat, cattle and pig, was viewed as a rapid, intentional process that started in one small region in southeastern Anatolia and then spread to many parts of Eurasia from there. Over the past quarter century, new research increasingly suggests that plant agriculture emerged gradually in many locations within the eastern Mediterranean region in response to local conditions. Although most researchers have embraced this new multiregional perspective, published narratives on animal domestication continue to highlight a core-periphery explanation. Here, I present evidence from my research projects in Central Anatolia, the southern Levant and Greece to build a case for a more gradual multiregional scenario for the emergence of sheep and goat domestication. The evidence emphasizes diverse pathways to animal domestication that were deeply influenced by local ecology and social conditions, and the transmission of information across large social networks. I conclude by discussing how ultimately, farming began to spread from Anatolia to Europe, making one of its first stops at Franchthi Cave on the coast of the Peloponnese in Greece.