Optical polishing methods and apparatus are disclosed for polishing and planarizing surfaces of a wafer or other workpiece. A polishing apparatus according to the present invention preferably comprises a dispenser for dispensing polishing agent between a polishing body and the surface of the object to be polished. The polishing apparatus moves the polishing body and the object being polished against one another. The polishing body preferably comprises a light-permeable substance. A film-thickness measuring device for measuring the thickness of a film on the object being polished is preferably positioned on the side of the polishing body opposite of the surface of the object being polished. The film-thickness measuring device preferably measures the film thickness by using a light source to irradiate the object being polished through the polishing body. The film-thickness measuring device then uses light reflected from the film of the object being polished to determine the film's thickness.