The offshore wind resource has huge energy potential. However, wind turbine floating structures have to withstand harsh conditions. Strong wind and wave effects combine to generate vibrations, fatigue, and heavy loads on the structure and other elements of the wind turbine. These structural problems increase maintenance requirements and risk of failure, while reducing availability and energy production. Another challenge for wind energy is to reduce production costs in order to be competitive with other alternatives. From the control point of view, the objective of lowering costs can be achieved by operating the turbine close to its optimum point of operation under partial load, guaranteeing reliability by reducing structural loads and regulating the power generated in strong wind regimes. In this typical and challenging context, this paper proposes a critical state-of-the-art review, discussing challenges and trends on floating offshore wind turbines control.