Responsible Artificial Intelligence (RAI) addresses the ethical and regulatory challenges of deploying AI systems in high-risk scenarios. This paper proposes a comprehensive framework for the design of an RAI system (RAIS) that integrates five key dimensions: domain definition, trustworthy AI design, auditability, accountability, and governance. Unlike prior work that treats these components in isolation, our proposal emphasizes their inter-dependencies and iterative feedback loops, enabling proactive and reactive accountability throughout the AI lifecycle. Beyond presenting the framework, we synthesize recent developments in global AI governance and analyze limitations in existing principles-based approaches, highlighting fragmentation, implementation gaps, and the need for participatory governance. The paper also identifies critical challenges and research directions for the RAIS framework, including sector-specific adaptation and operationalization, to support certification, post-deployment monitoring, and risk-based auditing. By bridging technical design and institutional responsibility, this work offers a practical blueprint for embedding responsibility throughout the AI lifecycle, enabling transparent, ethically aligned, and legally compliant AI-based systems.