An apparatus for cleaning surfaces of a printing press has an adjusting device having a pressing element for pressing a cleaning element against the surface to be cleaned when needed. The adjusting device has a drive for moving the pressing element relative to the surface to be cleaned between a home position, in which the cleaning element is positioned at a distance from the surface to be cleaned, and a cleaning position in which the cleaning element is in contact with the surface to be cleaned or, respectively, is in the direct proximity of the surface to be cleaned. An overload safety device is configured to respond upon a critical force acting on at least one of the adjusting device or pressing element being exceeded and to move the pressing element into an overload position.