The Romani, the largest European minority group with approximately 11 million people, constitute a mosaic of languages, religions, and lifestyles while sharing a distinct social heritage. Linguistic and genetic studies have located the Romani origins in the Indian subcontinent. However, a genome-wide perspective on Romani origins and population substructure, as well as a detailed reconstruction of their demographic history, has yet to be provided. Our analyses based on genome-wide data from 13 Romani groups collected across Europe suggest that the Romani diaspora constitutes a single initial founder population that originated in north/northwestern India ~1.5 thousand years ago (kya). Our results further indicate that after a rapid migration with moderate gene flow from the Near or Middle East, the European spread of the Romani people was via the Balkans starting ~0.9 kya. The strong population substructure and high levels of homozygosity we found in the European Romani are in line with genetic isolation as well as differential gene flow in time and space with non-Romani Europeans. Overall, our genome-wide study sheds new light on the origins and demographic history of European Romani.