Interest in floating offshore wind as a renewable energy source is growing, as it offers the potential to access deeper waters than those suited to bottom-fixed offshore turbines. A key design challenge for floating offshore wind turbines (FOWTs) is capturing the aerodynamic behaviour using numerical models, which is significantly more complex than for bottom fixed turbines due to the motions of the floating platform that result in unsteady relative wind flow at the rotor. Many of the engineering models available for analysing wind turbine aerodynamics such as the blade element momentum (BEM) method were designed for fixed turbines and require empirical corrections to account for unsteady aerodynamic effects, and may not be suitable for analysing the more complex aerodynamics associated with FOWTs. Higher order modelling approaches including computational fluid dynamics (CFD) may offer improved accuracy as they capture more of the flow physics, however, they can have extremely high associated computational costs. In this thesis, the performance of different aerodynamic models for FOWTs is investigated by studying the motion and load response of a FOWT in a range of load cases covering operational and extreme conditions using a BEM method and two different CFD-based models. Firstly, the BEM method used in the wind turbine engineering tool FAST is compared with an actuator line model (ALM) from the CFD wind turbine code package SOWFA for a range of load cases. Comparisons are made in load cases that have specific challenges for FOWTs and where the BEM method has known limitations, including rotor misalignment with the wind due to yaw, and varying wave conditions. The two modelling approaches are then used to study FOWT behaviour in realistic operational and extreme environmental conditions, and the model results are compared against available field data from full scale FOWT demonstration projects. The impact of using high order large eddy simulation (LES) to generate a turbulent wind field is also compared against a lower order statistical approach. Finally, a high order modelling approach is proposed that couples a geometry-resolved CFD model of a wind turbine blade with a structural model based on 3D finite element analysis (FEA) to enable two way coupled fluid structure interaction simulation. This model provides detailed information on the loading and deformation of blades, and is compared against FAST for studying a large flexible wind turbine blade in the parked and feathered position. This research provides improved understanding of the impact that the choice of aerodynamic and wind models have on the predicted response of a floating offshore wind turbine. ALM predictions are found to diverge from BEM predictions in increasingly large rotor yaw misalignment angles. Turbine loads and platform motions are found to be sensitive to the atmospheric stability condition, with stable conditions having a significant effect, however the use of high fidelity LES modelling of neutral conditions has little effect on turbine response (in either operational or extreme conditions) compared to using more efficient statistical modelling of turbulence using the Kaimal model. The results of the presented comparisons in this work are used to make recommendations on the use of different models in the design process for FOWTs. It is found that FAST is suitable for the majority of load cases, and may provide improved predictions of a FOWT in extreme conditions over an ALM that may underestimate aerodynamic loading on the tower. However, an ALM may provide improved predictions for a yawed turbine. The use of a high fidelity coupled CFD-FEA approach has potential to be a useful tool for analysing the detailed response of highly flexible blades where low fidelity methods are less reliable, though further work is needed to validate the modelling.