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The increasing deployment of robotic systems in complex and high-stakes environments, such as advanced manufacturing, healthcare, space exploration, and service robotics, requires robust strategies to ensure operational reliability, safety, and predictive maintenance. Real-time prognostics and health management, supported by recent advances in artificial intelligence, has emerged as a powerful approach for monitoring system health, detecting faults, and predicting failures before they occur. Unlike earlier review studies that mainly summarize traditional machine learning applications, the novelty of this paper lies in presenting a comprehensive taxonomy and critical synthesis of state-of-the-art AI-driven PHM techniques designed specifically for robotic systems. We evaluate a wide range of approaches, beginning with conventional machine learning models and extending to recent deep learning advancements, including transformers, vision transformers, and self-supervised learning frameworks. Furthermore, a novel contribution of this study is the rigorous benchmarking of their real-time feasibility, computational complexity, scalability, and performance trade-offs in practical robotic applications. In addition, this review introduces widely used benchmark datasets and highlights representative industrial case studies that demonstrate the practical effectiveness of AI-enabled PHM systems. The study also discusses important research gaps, including challenges related to model interpretability addressed through eXplainable AI, data privacy supported by federated learning, and the integration of cloud and edge computing within cloud robotics frameworks. Through a comprehensive gap matrix and quantitative comparative evaluations, this review provides insights to support the development of resilient, interpretable, and intelligent PHM systems for next-generation robotic applications.