The EU's Artificial Intelligence Act (AI Act) is a significant step towards responsible AI development, but lacks clear technical interpretation, making it difficult to assess models' compliance. This work presents COMPL-AI, a comprehensive framework consisting of (i) the first technical interpretation of the EU AI Act, translating its broad regulatory requirements into measurable technical requirements, with the focus on large language models (LLMs), and (ii) an open-source Act-centered benchmarking suite, based on thorough surveying and implementation of state-of-the-art LLM benchmarks. By evaluating 12 prominent LLMs in the context of COMPL-AI, we reveal shortcomings in existing models and benchmarks, particularly in areas like robustness, safety, diversity, and fairness. This work highlights the need for a shift in focus towards these aspects, encouraging balanced development of LLMs and more comprehensive regulation-aligned benchmarks. Simultaneously, COMPL-AI for the first time demonstrates the possibilities and difficulties of bringing the Act's obligations to a more concrete, technical level. As such, our work can serve as a useful first step towards having actionable recommendations for model providers, and contributes to ongoing efforts of the EU to enable application of the Act, such as the drafting of the GPAI Code of Practice.