We discuss superpolishing of a-cut sapphire which leads to a residual roughness less then 0.05 nm rms. There remains no damage layer. Polishing time is typically 15 to 30 minutes. The flatness is determined mainly by prepolishing. Curved surfaces can also be polished by this method. The lowest roughness is achieved, if removal rates are low, i.e. about 10 nm/min which corresponds to about half an atomic layer per second.