The service environment of offshore wind turbine (OWT) support structures is harsh, and it is extremely difficult to replace these structures during their operational lifespan, making their failure a catastrophic event. The structural health monitoring (SHM) of OWT support structures is a crucial aspect of operational maintenance for OWT support structures, aiming to mitigate significant financial losses. This paper systematically summarizes the current monitoring methods and technologies for OWT support structures, including towers and foundations. Through the review of monitoring content and the evolution of monitoring techniques for supporting structures, it delves deeper into the challenges faced by wind turbine monitoring and highlights potential avenues for future development. Then, the current damage identification techniques for OWT towers and foundations are analyzed, exploring various methods including model-based, vibration-based, artificial intelligence and hybrid fault diagnosis methods. The article also examines the advantages and disadvantages of each approach and outlines potential future directions for research and development in this field. Furthermore, it delves into the current damage identification techniques for OWT towers and foundations, discussing prevalent challenges and future directions in this domain. This status review can provide reference and guidance for the monitoring design of OWT support structures, and provide support for the fault diagnosis of OWT support structures.