In April 2021, the European Commission proposed a Regulation on Artificial Intelligence, known as the AI Act. We present an overview of the Act and analyse its implications, drawing on scholarship ranging from the study of contemporary AI practices to the structure of EU product safety regimes over the last four decades. Aspects of the AI Act, such as different rules for different risk-levels of AI, make sense. But we also find that some provisions of the draft AI Act have surprising legal implications, whilst others may be largely ineffective at achieving their stated goals. Several overarching aspects, including the enforcement regime and the effect of maximum harmonisation on the space for AI policy more generally, engender significant concern. These issues should be addressed as a priority in the legislative process.