Abstract: Machine for the abrasive machining of cylindrical parts, on which the parts to be machined are driven in rotation and in axial translational motion by supporting and driving means. The machine comprises a plurality of bands (17) of abrasive cloth having grains of increasing fineness in the direction of axial translational motion (A) of the parts to be machined, advancing means for unwinding each band (17) from a reel of new cloth continuously during the machining and for rewinding it onto a reel of worn cloth, bearing means for applying each band (17) with a predetermined pressure onto the parts to be machined, and oscillation means (18, 19, 20) for driving the said bearing means in an oscillating movement in axial translational motion.The invention is used, for example, for the superfinishing of shock-absorber rods, of cylindrical rollers, etc.