Wind energy industry is expanded to offshore and deep water sites, primarily due to the stronger and more consistent wind fields. Floating offshore wind turbine (FOWT) concepts involve new engineering and scientific challenges. A combination of waves, current, and wind loads impact the structures. Often under extreme cases, and sometimes in operational conditions, magnitudes of these loads are comparable with each other. The loads and responses may be large, and simultaneous consideration of the combined environmental loads on the response of the structure is essential. Moreover, FOWTs are often large structures and the load frequencies are comparable to the structural frequencies. This requires a fluid–structure–fluid elastic analysis which adds to the complexity of the problem. Here, we present a critical review of the existing approaches that are used to (i) estimate the hydrodynamic and aerodynamic loads on FOWTs, and (ii) to determine the structures’ motion and elastic responses due to the combined loads. Particular attention is given to the coupling of the loads and responses, assumptions made under each of the existing solution approaches, their limitations, and restrictions, where possible, suggestions are provided on areas where further studies are required.