Digital twins (DTs) are increasingly integrating Explainable Artificial Intelligence (XAI) to enhance trust, transparency, and informed decision-making across various application domains such as manufacturing, healthcare, energy systems, and transportation. As AI-driven DTs become more autonomous and complex, the need for interpretability becomes critical, particularly in safety-critical and regulatory-sensitive environments. This review explores the current state of the art in explainable intelligence within DTs, providing a taxonomy of XAI techniques based on model dependency, explanation format, and application context. It also surveys domain-specific implementations, including use cases in predictive maintenance, cybersecurity, battery health monitoring, and clinical diagnostics. Furthermore, the chapter discusses architectural considerations for embedding explainability into DT pipelines and outlines key challenges such as real-time interpretability, privacy concerns, evaluation inconsistency, and domain-specific generalizability. Finally, open research directions are identified to guide future efforts in building transparent, reliable, and human-centered digital twin systems.