The deployment of AI systems in the administration of justice raises a question that existing legal frameworks have yet to resolve: who bears responsibility for their outputs, and on what grounds. This book provides a systematic, socio-technical framework for distributing accountability across the entire AI value chain, from general-purpose model providers to deployers operating in high-stakes public decision-making contexts. Drawing on the COMPAS case and the emerging challenges posed by generative AI, the book analyses how AI's disruptive features and automation bias threaten human agency and create accountability gaps that existing legal frameworks cannot adequately address. It traces the theoretical foundations of algorithmic accountability by distinguishing between moral responsibility, accountability as a virtue, accountability as a mechanism, and civil liability - distinctions that prove decisive when the delegation of decision-making power to AI systems causes harm. Through analysis of the EU AI Act and its risk-based approach, the book identifies critical tensions between product safety mechanisms and fundamental rights protection, with particular attention to the concept of material influence as the threshold criterion for high-risk classification. It develops three accountability frameworks mapped onto the Act's risk classification structure: one for high-risk AI systems requiring mandatory human oversight; one for non-high-risk systems, where providers must demonstrate that their outputs do not materially influence judicial decisions; and one for limited-risk AI systems deployed by legal professionals. All three frameworks distribute responsibilities across the AI value chain and establish grounds for liability when automation bias causes harm. Central to the book's argument is the paradox of cognitive automation: as generative AI systems grow more persuasive, their opacity and tendency to 'hallucinate' undermine the very reliability that meaningful human oversight presupposes. The book argues that neither technical solutions nor formal oversight mandates can adequately address these risks without integrating cognitive biases and human factors into accountability design; a socio-technical approach is therefore essential to any credible framework for Responsible AI. Practical proposals for contestability are developed through analysis of the right to explanation and liability regimes, situating ex ante risk management and ex post redress mechanisms within a coherent governance architecture.