Floating wind turbines enable harvesting the offshore wind resources over deep sea. About 20 concepts are under development, at varying stages of maturity. Two concepts are demonstrated in full scale; these are HyWind and WindFloat. Both employ a standard on-shore wind turbine with only minor modifications, but on a spar and a semi-submersible floater, respectively. Other concepts suggest new types of turbines, e.g., the DeepWind concept consisting of a vertical axis turbine and a subsea generator. The three concepts represent different approaches: HyWind and WindFloat are already in a demonstration phase applying (mostly) well-known technology, albeit in a new setting. DeepWind is a European research project based mostly on new technology. The concepts are described in some detail with emphasis on control and operation. Prospects are discussed including technical challenges and a performance metric of energy production per unit steel mass. Floating offshore wind turbines represent a promising technology. The successful operation of HyWind and WindFloat in full scale demonstrates a well advanced technology readiness level, where further development will go into refining the concepts, cost reductions, and series production for installation of wind farms. DeepWind is in an early phase and not proven, but with promising perspectives.