A global optimization tool has been developed that enables identification and evaluation of optimal support structure configurations for floating offshore wind turbines. Floating support structures have the potential to dramatically expand the domain of the offshore wind energy industry by providing access to deep-water sites that are out of reach of conventional bottom-fixed support structures. The technology is young, and the complex floating platform design problem inhibits a convergence of designs. Researchers have tried using geometry parameterization techniques to identify optimal points in the floating platform design space. However, these past efforts have been limited in breadth, suffering from parameterizations that exclude a number of important design options. The present work explores the floating platform design space more fully by applying a flexible, genetic algorithm-based optimization framework. Judiciously-selected decision variables drive a geometry model that supports a diverse range of floating platform configurations and an accompanying mass model that features intelligent ballast distribution. The panel method code WAMIT, supplemented by a custom viscous drag model, provides platform hydrodynamic characteristics. A coupled frequency-domain floating wind turbine model is used to evaluate the system dynamics, which in turn inform the objective function. The exploratory design optimization made possible by this framework is shown to yield a set of distinct locally-optimal designs – some that are similar to well-established spar and semi-submersible configurations, and some that are less conventional. This broader exploration of the design space is therefore useful in both verifying the optimality of existing design concepts and identifying promising configurations that lie outside existing intuition-guided design concepts.