This paper explores the potential complementarity and overlap between the Artificial Intelligence Act (AI Act) and the General Data Protection Regulation (GDPR), explicitly examining safeguarding individual interests in the context of explainable AI systems. The authors make two novel contributions. Firstly, the paper comprehensively outlines the rules for explainable AI as stipulated in the GDPR and the AI Act. Secondly, it delves into the interaction between these legal acts, thereby analysing the types of explanations recognised under both the AI Act and the GDPR, the interpretation of legal requirements, and the implications for high-risk AI systems through the lens of a right to an explanation. Additionally, this paper examines other emerging frameworks in the European Union (EU) that focus on protecting individuals while integrating AI systems into the EU market. The development and deployment of AI systems raise significant concerns regarding the necessity for additional regulatory frameworks within the EU's legal landscape. Consequently, this paper posits that legal requirements for explainable AI likely extend beyond the current GDPR and AI Act regulations, necessitating a broader application of legal rules relevant to a particular sector.