The invention provides methods of polishing a substrate comprising (i) contacting a substrate comprising at least one metal layer comprising copper with a chemical-mechanical polishing (CMP) system and (ii) abrading at least a portion of the metal layer comprising copper to polish the substrate. The CMP system comprises (a) an abrasive, (b) an amphiphilic nonionic surfactant, (c) a means for oxidizing the metal layer, (d) an organic acid, (e) a corrosion inhibitor, and (f) a liquid carrier. The invention further provides a two-step method of polishing a substrate comprising a first metal layer and a second, different metal layer. The first metal layer is polishing with a first CMP system comprising an abrasive and a liquid carrier, and the second metal layer is polished with a second CMP system comprising (a) an abrasive, (b) an amphiphilic nonionic surfactant, and (c) a liquid carrier.