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The paper deals with the optimum design of mooring systems, with chain cable catenary line configuration, for floating offshore wind turbines with tri-floater support structure in deep waters. Operational and extreme climate conditions are assessed for a candidate location in southern Tyrrhenian Sea, determining wind, wave and current loads. The platform response is assessed in terms of staric loads due to turbine thrust, wind, current and steady drift forces. The platform dynamic offset is determined considering both wave-frequency linear and low-frequency second-order motions. Two different mooring systems with 9 (equally spaced) and 11 (non-uniformly spaced) lines are selected in order to minimize the total mass; the incidence of both line number and pattern is appreciable and there is wide space to achieve considerable weight reductions, so increasing the overall wind farm cost-performance trade-off