Procurement is the gateway for public sector AI adoption, as well as an area of active experimentation with AI. Both 'AI procurement' and 'AI in procurement' are giving rise to emerging regulatory and governance strategies and practices that shed light on the challenges and constraints of current approaches to 'digital transformation'. This contribution explores emerging trends and identifies areas for future research, with a specific focus on the promotion of responsible and trustworthy AI adoption. In relation to AI procurement, it discusses 'contractbased' and 'standard-based' approaches to its governance, as well as the emerging pivot towards prescriptive AI procurement legislation. In relation to AI in procurement, it explores the emergence of safe harbours and persistent challenges for AI use by public buyers, and maps some of the challenges arising from bidders' (improper) AI use. It concludes by bringing together themes from the two related issues of procuring AI and using AI in procurement.