Floating offshore wind turbines have the potential to become a significant source of affordable renewable energy. However, their strong interactions with both wind- and wave-induced forces raise a number of technical challenges in both modelling and design. This thesis takes aim at some of those challenges. One of the most uncertain modelling areas is the mooring line dynamics, for which quasi-static models that neglect hydrodynamic forces and mooring line inertia are commonly used. The consequences of using these quasi-static mooring line models as opposed to physically-realistic dynamic mooring line models was studied through a suite of comparison tests performed on three floating turbine designs using test cases incorporating both steady and stochastic wind and wave conditions. To perform this comparison, a dynamic finite-element mooring line model was coupled to the floating wind turbine simulator FAST. The results of the comparison study indicate the need for higher-fidelity dynamic mooring models for all but the most stable support structure configurations. Industry consensus on an optimal floating wind turbine configuration is inhibited by the complex support structure design problem; it is difficult to parameterize the full range of design options and intuitive tools for navigating the design space are lacking. The notion of an alternative, "hydrodynamics-based'' optimization approach, which would abstract details of the platform geometry and deal instead with hydrodynamic performance coefficients, was proposed as a way to obtain a more extensive and intuitive exploration of the design space. A basis function approach, which represents the design space by linearly combining the hydrodynamic performance coefficients of a diverse set of basis platform geometries, was developed as the most straightforward means to that end. Candidate designs were evaluated in the frequency domain using linearized coefficients for the wind turbine, platform, and mooring system dynamics, with the platform hydrodynamic coefficients calculated according to linear hydrodynamic theory. Results obtained for two mooring systems demonstrate that the approach captures the basic nature of the design space, but further investigation revealed limitations on the physical interpretability of linearly-combined basis platform coefficients. A different approach was then taken for exploring the design space: a genetic algorithm-based optimization framework. Using a nine-variable support structure parameterization, this framework is able to span a greater extent of the design space than previous approaches in the literature. With a frequency-domain dynamics model that includes linearized viscous drag forces on the structure and linearized mooring forces, it provides a good treatment of the important physical considerations while still being computationally efficient. The genetic algorithm optimization approach provides a unique ability to visualize the design space. Application of the framework to a hypothetical scenario demonstrates the framework's effectiveness and identifies multiple local optima in the design space -- some of conventional configurations and others more unusual ... .