Tank system of a motor vehicle with a volume-modifying element provided in the interior of the fuel tank at least substantially above the liquid level, which volume-modifying element forms a compensating volume which is connected to the surroundings, wherein the volume-modifying element and a tank ventilation line are conceived in such a manner that the vapour pressure equilibrium in the fuel tank is able to be maintained by automatically supplying and discharging ambient air into or out of the volume-modifying element to such an extent that the tank ventilation line can remain closed, and having a valve arrangement which is assigned to the tank ventilation line, characterized in that the valve arrangement closes in a float-controlled manner when filling the fuel tank and thus brings about a maximum fill level in the fuel tank, and in that an influence of the volume-modifying element on the maximum fill level is eliminated in that a) the volume-modifying element is arranged in the fuel tank in such a manner that in the case of a vehicle standing on a horizontal surface, as seen in the vertical axis direction, the upper side of the volume-modifying element is not located appreciably above that fuel fill level at which the stated valve arrangement closes; and/or b) there is provided in the connection between the compensating volume of the volume-modifying element and the surroundings a valve which, when the fill level increasing during filling has reached the maximum fill level, is actuated in the closing direction, or is at least partially closed by a float which is displaced as a result of the fill level which rises during filling. wherein the volume-modifying element and a tank ventilation line are conceived in such a manner that the vapour pressure equilibrium in the fuel tank is able to be maintained by automatically supplying and discharging ambient air into or out of the volume-modifying element to such an extent that the tank ventilation line can remain closed, and having a valve arrangement which is assigned to the tank ventilation line, characterized in that the valve arrangement closes in a float-controlled manner when filling the fuel tank and thus brings about a maximum fill level in the fuel tank, and in that an influence of the volume-modifying element on the maximum fill level is eliminated in that a) the volume-modifying element is arranged in the fuel tank in such a manner that in the case of a vehicle standing on a horizontal surface, as seen in the vertical axis direction, the upper side of the volume-modifying element is not located appreciably above that fuel fill level at which the stated valve arrangement closes; and/or b) there is provided in the connection between the compensating volume of the volume-modifying element and the surroundings a valve which, when the fill level increasing during filling has reached the maximum fill level, is actuated in the closing direction, or is at least partially closed by a float which is displaced as a result of the fill level which rises during filling. a) the volume-modifying element is arranged in the fuel tank in such a manner that in the case of a vehicle standing on a horizontal surface, as seen in the vertical axis direction, the upper side of the volume-modifying element is not located appreciably above that fuel fill level at which the stated valve arrangement closes; and/or b) there is provided in the connection between the compensating volume of the volume-modifying element and the surroundings a valve which, when the fill level increasing during filling has reached the maximum fill level, is actuated in the closing direction, or is at least partially closed by a float which is displaced as a result of the fill level which rises during filling.