Tension leg platform (TLP) floating wind turbines (FWTs) represent a promising frontier in the research and development of the offshore wind industry. The recent completion of the first TLP floating wind farm, Provence Grand Large, marks a significant milestone. Compared to other FWT designs, TLP FWTs have advantages in motion dynamics and structural utilization, leading to improved performance and reduced cost. However, their unique physical characteristics require specialized design considerations and innovative solutions for practical implementation. To facilitate the development of next-generation TLP FWTs, this paper presents a state-of-the-art review focused on various design aspects: development, technical challenges, requirements, innovative concepts, and research methods. An overview is provided, detailing key design parameters of existing TLP FWTs, such as weight, displacement, installed capacity, and the natural period of motion. Additionally, the design challenges of TLP FWTs are summarized. Afterwards, the TLP FWT designs are classified and critically reviewed, highlighting the design features and installation methods of innovative concepts while identifying development trends in the field. Then, numerical and experimental methods for design are discussed, along with current research gaps. Finally, this paper presents four key areas for future research on TLP FWTs.