Prognostics and health management (PHM) enables proactive maintenance and risk mitigation by monitoring, analyzing, and forecasting equipment health, undergirding safe and stable system operation. While physics-based and data-driven PHM paradigms are relatively mature, classical approaches struggle to integrate expert knowledge and generalize when confronted with massive heterogeneous data, especially unstructured text and multimodal information generated by increasingly complex industrial systems. Recently, the rapid development of Transformer architecture and large language model (LLM) have opened a high-precision prediction paradigm that efficiently exploits domain expertise, offering advantages such as knowledge extraction and fusion, few-shot generalization, and intelligent decision support. This review comprehensively surveys the current status and future prospects of PHM empowered by LLM. First, canonical PHM tasks, together with Transformer architecture and general LLM, are introduced. Second, domain-specific LLM construction is elaborated with respect to domain-knowledge creation and injection, covering internal parameter optimization and external knowledge augmentation; a unified framework for PHM domain-specific LLM is presented. Third, current PHM domain-specific LLM-based frameworks and applications are dissected in-depth from a task-oriented perspective across components, subsystems, and complex systems levels, focusing on fault diagnosis, state of health estimation, remaining useful life prediction, and anomaly detection. Finally, future challenges and opportunities are outlined regarding model compression, edge deployment, and generalized complex systems.