The present invention relates to a method for the electrochemical polishing of surfaces of metals and metal alloys, in particular surfaces of niobium, niobium alloys, tantalum and tantalum alloys. It uses an electrolyte that comprises methanesulfonic acid and ammonium bifluoride, and which is also an object of the present invention. Said electrolyte makes it possible to electropolish niobium-containing and/or tantalum-containing workpieces efficiently, with high quality and in a manner that is harmless both for the workpiece and for humans and the environment.