As wind turbines get larger in size and advance into deep waters, the Floating Wind Turbine (FWT) emerges as a promising alternative technique in the wind power community. Under the combined excitations of external environmental loads and internal structure vibrations, the floating platform of FWT tends to experience complex random motions. It will affect the dynamic stability of the FWT system, increase the structural fatigue loads, and threaten the safety of the whole FWT system. Therefore, it is of great significance to develop reliable and controllable methods to suppress the platform motions. The major methods and techniques of FWT motion suppression are introduced and analyzed, in terms of the basic principles, technical characteristics, research status and application situations. It is expected to serve as a reference for researchers in related fields.