Abstract: The apparatus for grinding of the inner or the outer surface of an annular workpiece, especially a race ring for an antifriction bearing, includes a fixed holder for hydrodynamically centering the workpiece, a rotating driver, and a pressing device. A respective workpiece can be held by the holder and is pressed with one end face by the pressing device against the respective end face of the driver for coaxially rotating the workpiece and the driver. The pressing device includes a hydraulic cyliner/piston assembly, which can be pressed against the workpiece on the side which is opposite to the driver. The pressing device includes a pressing ring which is formed as an extension of the piston and which, via the respective workpiece, is rotatively driven by the driver. The pressing piston is arranged in a cylinder which is disposed coaxially with respect to the driver, but it is mounted fixedly in the machine so as not to rotate.