Abstract: A machine for machining internal surfaces of axisymmetric workpieces with an abrasive strip. The machine includes a roller for pressing the abrasive strip against a surface of the workpiece that is to be machined. The workpiece rotates about an axis, and the roller rotates about an axis parallel to the axis of the workpiece. The machine also includes a pulley positioned on either side of the roller, for guiding the abrasive strip to the roller and holding the abrasive strip in position relative to the roller. The abrasive is driven continuously over the roller during machining, to renew the abrasive where the roller presses the abrasive strip against the surface of the workpiece. The roller and pulleys can be moved oointo, out of, and within a workpiece.