A polishing apparatus and method has a function of polishing a surface of a film formed on a substrate to a flat mirror finish and a function of polishing unnecessary metal film such as copper film deposited on an outer peripheral portion of the substrate to remove such unnecessary metal film. The polishing apparatus comprises a surface polishing mechanism comprising a polishing table having a polishing surface and a top ring for holding the substrate and pressing the substrate against the polishing surface of the polishing table to thereby polish a surface of the substrate, and an outer periphery polishing mechanism for polishing an outer peripheral portion of the substrate.