A polishing is used for polishing a substrate such as a semiconductor wafer to a flat mirror finish. The polishing apparatus includes a polishing table having a polishing surface, a substrate holding apparatus configured to hold the substrate and to press the substrate against the polishing surface, and a controller. The substrate holding apparatus includes an elastic membrane configured to forma substrate holding surface which is brought into contact with the substrate, a carrier provided above the elastic membrane, at least one pressure chamber formed between the elastic membrane and the carrier, and an infrared light detector configured to measure thermal energy from the elastic membrane. The controller calculates an estimate value of a temperature of the elastic membrane using a measured value of the infrared light detector.