An anode of a rotary-anode X-ray tube is connected to a bearing portion which is rotatable about an axis of rotation and which cooperates with the stationary bearing portion in which there is provided a cavity which extends in the direction of the axis of rotation and whose side walls can be cooled by means of a cooling medium. Effective cooling, in combination with a small pressure drop of the cooling medium, is achieved in that there is provided a cooling member which serves to produce an essentially laminar cooling medium flow and which consists of a number of lamellae which extend essentially parallel to the axis and which are in thermal contact with the side walls of the cavity.