An offshore wind turbine installation vessel (1) for installation of an offshore wind turbine (500), wherein the wind turbine is of the type to be installed on a foundation (600) 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 (501) to be fitted with its lower end onto the foundation, and with a nacelle with a hub and blades supported on top of said mast, wherein said vessel comprises: a hull (1a, 1b); preferably a non-jack-up type floating hull; preferably a self-propelled floating hull; a crane structure (200) extending upward from said hull; wherein the crane structure is provided with a hoisting device having one or more wind turbine suspension elements (400) and a wind turbine engagement device (450) supported by said one or more suspension elements and adapted to engage with said wind turbine, said hoisting device being adapted to support and to raise and lower in controllable manner at least the mast of the wind turbine while in vertical orientation; preferably with the nacelle and preferably also with the hub and blades fitted on top of the mast.