The wafer polishing apparatus comprises a polishing plate, a polishing head capable of holding a wafer, and a slurry supplying section. The polishing plate includes: a plurality of concentric polishing zones, each of which has a prescribed width for polishing the wafer and on each of which a polishing cloth is adhered; and a groove for discharging slurry being formed between the polishing zones. A head cleaning section, which cleans the polishing head, or a wafer cleaning section, which cleans the polished wafer, is provided to a center part of the polishing plate and located on the inner side of the innermost polishing zone.