With the accelerating availability of multimodal surgical data and real-time computation, Surgical Digital Twins (SDTs) have emerged as virtual counterparts that mirror, predict, and inform decisions across pre-, intra-, and postoperative care. Despite promising demonstrations, SDTs face persistent challenges: fusing heterogeneous imaging, kinematics, and physiology under strict latency budgets; balancing model fidelity with computational efficiency; ensuring robustness, interpretability, and calibrated uncertainty; and achieving interoperability, privacy, and regulatory compliance in clinical environments. This survey offers a critical, structured review of SDTs. We clarify terminology and scope, propose a taxonomy by purpose, model fidelity, and data sources, and synthesize state-of-the-art achievements in deformable registration and tracking, real-time simulation and co-simulation, AR/VR guidance, edge-cloud orchestration, and AI for scene understanding and prediction. We contrast non-robotic twins with robot-in-the-loop architectures for shared control and autonomy, and identify open problems in validation and benchmarking, safety assurance and human factors, lifecycle "digital thread" integration, and scalable data governance. We conclude with a research agenda toward trustworthy, standards-aligned SDTs that deliver measurable clinical benefit. By unifying vocabulary, organizing capabilities, and highlighting gaps, this work aims to guide SDT design and deployment and catalyze translation from laboratory prototypes to routine surgical care.