The present invention is directed to a CMP polishing slurry comprising cerium oxide particles, an organic compound having an acetylene bond (triple bond between carbon and carbon) and water, and a method for polishing a substrate which comprises a step of polishing a film to be polished of the substrate with the polishing slurry. In a CMP (chemical mechanical polishing) technique for flattening inter layer dielectrics, insulating films for shallow trench isolation and the like in a manufacturing process of semiconductor devices, the present invention enables the effective and high-speed polishing.