A wafer polishing apparatus has a base and a turntable having a polishing pad thereon and mounted on the base for rotation of the turntable and polishing pad relative to the base about an axis perpendicular to the turntable and polishing pad. The polishing pad includes a work surface engageable with a front surface of a wafer for polishing the front surface of the wafer. A drive mechanism is mounted on the base for imparting rotational motion about an axis substantially parallel to the axis of the turntable. A polishing head is connected to the drive mechanism for driving rotation of the polishing head. The polishing head has a pressure plate adapted to hold the wafer for engaging the front surface of the wafer with the work surface of the polishing pad. The pressure plate has a generally planar position and is selectively movable from the planar position to a convex position and to a concave position.