This paper describes the minimum-cost design of spar-type floating support structures and mooring system for offshore wind as an optimization problem: Find the cheapest solution that satisfies a set of design requirements. The present approach utilizes available response analysis programs for mooring system forces and vessel motions, and combines this with a gradient search method for solution of nonlinearoptimization problems with arbitrary constraints. Design constraints considered are: Vessel motion, tower inclination, tower top acceleration, spar draught, mooring line load limitations, minimum horizontal pretension, and maximum horizontaloffset. The spar buoy is modelled as composed of a set of cylindrical sections with different mass, buoyancy and cost properties, where each section is assumed to have a uniform mass distribution.It is shown that acceptable solutions can be found even when starting from infeasible startingpoints. A parameter study is performed and the results demonstrates the relative importanceof some of the key design parameters, mainly the shape and weight distribution of the spar buoy structure. Three different spar configurations are considered. Variations of design parameters include turbine mass, maximum heel angle, and maximum draught.