Artificial intelligence (AI) is increasingly embedded in the design, operation and oversight of financial market infrastructures (FMIs), including payment systems, central counterparties, securities settlement systems and collateral frameworks. These applications range from fraud detection and operational monitoring to risk measurement, margin setting and the operationalization of public policy. While AI offers the potential to enhance efficiency, resilience and financial stability, its use within FMIs raises unique and heightened concerns, given their systemic importance and public-good characteristics. This paper examines the current and prospective role of AI in FMIs, analyzes key risks and opportunities, and it develops a principled framework for the responsible use of AI based on the principles of explainability, data stewardship, governance and ethics (EDGE). It argues that responsibility is not a constraint on innovation but a necessary condition for trust, legitimacy and the sustainable use of AI in FMIs.