Floating offshore wind is a promising renewable energy source for several Mediterranean Countries. The exploitation of this resource will contribute to reducing carbon dependence and support the clean energy transition towards a climate neutral Europe. This work presents a novel methodology for estimating spatially-resolved Levelised Cost of Energy and offshore wind energy potential to provide optimal design of floating offshore wind farms in the Mediterranean Sea. For the first time, each site is optimised based on electrical grid cable design and wind farm layout optimisation using the Jensen wake model. The largest technical capacity potentials are obtained in Libya, Tunisia, Italy and Greece, accounting for 72.2 % of the total Mediterranean potential with a total installed capacity of 782 GW. The average LCOE is 93.4 €/MWh and the average capacity factor is 31.8 %, while 67.5 % of the technical potential has LCOE below 90 €/MWh which demonstrates that floating offshore wind in the Mediterranean could become soon competitive with other renewable energies. Optimal floating wind farm design parameters show the prevalence of a wind farm array of 10x10 wind turbines with a preferred rated power of 15 MW and the HVDC export cable connection. Among the selected floating platforms, Hywind outperforms WindFloat and GICON-SOF in 59.2 % of the suitable areas due to the lower structure material. Policymakers and stakeholders will primarily benefit from this study, which provides them with important information for careful marine spatial planning and the development of floating offshore wind farms in the Mediterranean.