This thesis is aimed at reliable numerical predictions of hydrodynamic damping and loads on floating offshore wind turbines. Investigations of hydrodynamic damping and loads on a moored semi-submersible floater are presented using Reynolds-Averaged Navier-Stokes equations. The investigations include systematic verification studies comparing three discretisation error and uncertainty quantification methods on four different aspects of floating wind turbine simulations. These aspects are wave propagation, wave loads on a cylinder, surge decay motion of a semi-submersible, and a dynamic mooring model in surge decay motion. The discretisation error and uncertainty estimation approaches are based on Richardson extrapolation. Furthermore, a variety of different settings and models are studied to predict hydrodynamic damping, including temporal and spatial discretisation, wave radiation as well as turbulence and mooring models. For each model and numerical setting, the hydrodynamic damping in terms of the linear and quadratic component along with the flow field is analysed.