The indigenous populations of inner Eurasia-a huge geographic region covering the central Eurasian steppe and the northern Eurasian taiga and tundra-harbour tremendous diversity in their genes, cultures and languages. In this study, we report novel genome-wide data for 763 individuals from Armenia, Georgia, Kazakhstan, Moldova, Mongolia, Russia, Tajikistan, Ukraine and Uzbekistan. We furthermore report additional damage-reduced genome-wide data of two previously published individuals from the Eneolithic Botai culture in Kazakhstan (~5,400 BP). We find that present-day inner Eurasian populations are structured into three distinct admixture clines stretching between various western and eastern Eurasian ancestries, mirroring geography. The Botai and more recent ancient genomes from Siberia show a decrease in contributions from so-called 'ancient North Eurasian' ancestry over time, which is detectable only in the northern-most 'forest-tundra' cline. The intermediate 'steppe-forest' cline descends from the Late Bronze Age steppe ancestries, while the 'southern steppe' cline further to the south shows a strong West/South Asian influence. Ancient genomes suggest a northward spread of the southern steppe cline in Central Asia during the first millennium BC. Finally, the genetic structure of Caucasus populations highlights a role of the Caucasus Mountains as a barrier to gene flow and suggests a post-Neolithic gene flow into North Caucasus populations from the steppe.