The proposed European Artificial Intelligence Act (AIA) is the first attempt to elaborate a general legal framework for AI carried out by any major global economy. As such, the AIA is likely to become a point of reference in the larger discourse on how AI systems can (and should) be regulated. In this article, we describe and discuss the two primary enforcement mechanisms proposed in the AIA: the conformity assessments that providers of high-risk AI systems are expected to conduct, and the post-market monitoring plans that providers must establish to document the performance of high-risk AI systems throughout their lifetimes. We argue that AIA can be interpreted as a proposal to establish a Europe-wide ecosystem for conducting AI auditing, albeit in other words. Our analysis offers two main contributions. First, by describing the enforcement mechanisms included in the AIA in terminology borrowed from existing literature on AI auditing, we help providers of AI systems understand how they can prove adherence to the requirements set out in the AIA in practice. Second, by examining the AIA from an auditing perspective, we seek to provide transferable lessons from previous research about how to refine further the regulatory approach outlined in the AIA. We conclude by highlighting seven aspects of the AIA where amendments (or simply clarifications) would be helpful. These include, above all, the need to translate vague concepts into verifiable criteria and to strengthen the institutional safeguards concerning conformity assessments based on internal checks.