A damping device for damping or avoiding pressure surges, such as pulses, in hydraulic supply circuits has a damping housing (1) surrounding a damping chamber and having a fluid inlet (35) and a fluid outlet (41). A fluid receiving chamber extends between the fluid inlet (35) and the fluid outlet (41). During operation of the device, a fluid flow crosses the damping chamber in a throughflow direction (11), from the fluid inlet (35) to the fluid outlet (41). Parts of the fluid receiving chamber extend transversely with respect to the throughflow direction (11). More than one fluid receiving chamber is arranged one after the other in the throughflow direction (11). The fluid receiving chamber that is first upstream and the fluid receiving chamber that is last downstream immediately adjoin the fluid inlet (35) and the fluid outlet (41), respectively.