The Paper aims to explore whether the current ecosystem of existing and still-to-be adopted rules on artificial intelligence (AI) systems in the European Union does fully and adequately address the liability for damages caused by Generative AI system. It maps first and primarily the distinctive features and functional characteristics of Generative AI likely to impact on regulatory and legal considerations and, in particular, on determining the specific regulatory regime governing Generative AI and on tracing and allocating liability along the value chain pursuant to the AI Act. On the basis of this mapping exercise, the Paper focuses on testing the liability rules as provided for the draft Artificial Intelligence Liability Directive and the Revised Product Liability Directive and assessing their sufficiency and effectiveness in the face of Generative AI. Beyond the assessment of the rules laid down in the above-referred texts, the Paper briefly describes other liability scenarios to be explored in future works.