The invention relates to an acidic, aqueous, chromium-free composition (A) for the anti-corrosive treatment of steel and/or galvanized steel surfaces comprising metal ions (M) selected from ions at least of the elements nickel, cobalt, molybdenum, iron or tin and a multi-stage method applying the composition (A) for the anti-corrosive pre-treatment of metal components which have steel and/or galvanized steel surfaces. The invention further relates to metal surfaces of zinc or iron having a passive layer system comprising at least 30 mg/m2 nickel and at least 10 mg/m2 zircon, titanium and/or hafnium and sulfur, wherein nickel is present in metallic form at up to at least 30 At. %, obtainable in a method according to the invention.