Modern power systems are evolving due to increasing penetration of renewable energy sources, deeper participation of the demand side, and widespread deployment of advanced information and digital technologies. As a result, system operation and control are becoming increasingly challenging. Large language models (LLMs), with their advanced capabilities in semantic understanding and knowledge reasoning, offer a promising tool to support the operation, control, analysis, and decision-making of power systems. This paper provides a comprehensive review of LLM applications in power systems, encompassing four representative application domains: power grid, power equipment, demand side, and electricity market and policy-making. Based on the functional roles and implementations of LLMs, four major application strategies are identified: model adaptation, capability enhancement, multimodality integration, and multi-agent coordination. In addition, the core functions, representative methods, and evolving trends of LLM applications are reviewed across different domains. Finally, key challenges in applying LLMs to power systems are discussed, and future research directions are outlined with regard to ensuring physical feasibility, enhancing data efficiency and privacy, and improving interpretability and rationality.