A generator for a wind turbine including a rotor arrangement and a stator arrangement is provided. The rotor arrangement includes a cylinder barrel or a sleeve enclosing a rotor element provided for the production of energy and having an outer cylinder barrel shaped surface in relation to a centre axis of the generator acting as a bearing surface of the rotor arrangement. The stator arrangement includes a cylinder barrel or a sleeve covering a stator element provided for the production of energy and having an inner cylinder barrel shaped surface in relation to the centre axis of the generator acting as a bearing surface of the stator arrangement. The bearing surfaces of the rotor arrangement and the stator arrangement are arranged oppositely to each other with a substantially cylinder barrel shaped air gap in-between, wherein the air gap comprises a lubricant. Also, a wind turbine including a generator is provided.