Wet-running clutch unit for a motor vehicle comprising a friction clutch with at least one first friction disc which is assigned to a drive element and at least one second friction disc which is assigned to an output element, an alternating arrangement of the first friction disc and the second friction disc forming a friction disc assembly which can be brought into a frictional connection, and a fluid distributor, the fluid distributor being arranged fixedly on the drive element and having at least one fluid chamber, the first friction disc having at least one first recess and the second friction disc having at least one second recess), the first recess of the first friction disc and the second recess of the second friction disc overlapping in the radial and axial direction, with the result that at least one friction disc assembly recess is formed, the fluid chamber of the fluid distributor being configured in such a way that it extends in the axial direction into the friction disc assembly recess.