An offshore wind turbine installation vessel, wherein the wind turbine is of the type to be installed on a foundation that is installed on the seabed prior to the installation of the wind turbine on the foundation, wherein the wind turbine is of the type with a vertical mast to be fitted with its lower end onto the foundation, wherein the vessel includes a non-jack-up type floating hull; and a crane structure extending upward from said hull. The crane structure is provided with a hoisting device having one or more wind turbine suspension elements and a wind turbine engagement device supported by the suspension elements and adapted to engage with said wind turbine. The hoisting device is adapted to support and to raise and lower in controllable manner at least the mast of the wind turbine while in vertical orientation.