Governance efforts for artificial intelligence (AI) are taking on increasingly more concrete forms, drawing on a variety of approaches and instruments from hard regulation to standardisation efforts, aimed at mitigating challenges from high-risk AI systems. To implement these and other efforts, new institutions will need to be established on a national and international level. This paper sketches a blueprint of such institutions, and conducts in-depth investigations of three key components of any future AI governance institutions, exploring benefits and associated drawbacks: (1) “purpose”, relating to the institution’s overall goals and scope of work or mandate; (2) “geography”, relating to questions of participation and the reach of jurisdiction; and (3) “capacity”, the infrastructural and human make-up of the institution. Subsequently, the paper highlights noteworthy aspects of various institutional roles specifically around questions of institutional purpose, and frames what these could look like in practice, by placing these debates in a European context and proposing different iterations of a European AI Agency. Finally, conclusions and future research directions are proposed.