A hydrodynamic, non-contacting end-face seal for a rotary machine comprises a rotor mounted on the machine shaft and a ring-shaped stator facing the flat and smooth surface of the rotor at a short distance. The stator encloses the machine shaft, is eccentrically positioned in respect of the rotor center, and is urged towards the rotor surface by a helical spring. The surface of the stator ring is shaped to form a circular rim separated from the rotor surface by a very small gap of varying width, and the space enclosed by this rim is raised to a higher pressure than the surroundings by the hydrodynamic action of the surface velocity of the rotor on the liquid in the machine, by making the gap width as follows: the gap width between rotor and stator is of a mechanically permissible minimum in all points of the rim at which the velocity vector of the rotor surface extends from the inside of the rim to the outside, and the gap width is larger in all points of the rim at which the velocity vector extends from the outside to the inside. The gap width is changed, in accordance with pressure conditions in the machine by action of the helical sprin