The European Union has proposed the Artificial Intelligence Act intending to regulate AI systems, especially those used in high-risk, safety-critical applications such as healthcare. Among the Act's articles are detailed requirements for transparency and explainability. The field of explainable AI (XAI) offers technologies that could address many of these requirements. However, there are significant differences between the solutions offered by XAI and the requirements of the AI Act, for instance, the lack of an explicit definition of transparency. We argue that collaboration is essential between lawyers and XAI researchers to address these differences. To establish common ground, we give an overview of XAI and its legal relevance followed by a reading of the transparency and explainability requirements of the AI Act and the related General Data Protection Regulation (GDPR). We then discuss four main topics where the differences could induce issues. Specifically, the legal status of XAI, the lack of a definition of transparency, issues around conformity assessments, and the use of XAI for dataset-related transparency. We hope that increased clarity will promote interdisciplinary research between the law and XAI and support the creation of a sustainable regulation that fosters responsible innovation.