Mooring systems are essential for ensuring the safety and station-keeping of floating offshore structures. Despite advances in mooring technologies over the past two decades, several design challenges remain. The design of mooring systems is a multidisciplinary task that requires expertise in marine hydrodynamics, structural mechanics, materials engineering, and mechanical design. This paper reviews state-of-the-art mooring designs for floating wind turbines, addressing key aspects such as floater-mooring interactions, material and component selection, design methods and guidelines, simulation tools, and innovative solutions. This review critically examines existing mooring designs for three types of floating wind turbines from 20 projects, presents eight mooring materials, and compares three design guidelines. Additionally, three innovative mooring solutions are highlighted at the farm level. Finally, six areas for future research are identified. This review aims to contribute to the development of safe, robust, and cost-effective mooring systems for the floating wind industry.