Abstract: The invention relates to a method for coating metal surfaces with an acidic aqueous conversion composition which contains: in total 0.01 to 1 g/l of TiF62+, ZrF62+ and/or HfF62 calculated as ZrF62+, 0 or 0.01 to 1 g/l in each case of Fe2+, Mn and/or Zn ions, of which at least one type of these ions is present in a content range from 0.01 to 1 g/l, 0 or 0.01 to 2 g/l of organic polymer and/or copolymer, 0 or 0.01 to 2 g/l of ultrafme particulate SiO2, approximately 0 or 0.01 to 10 g/l of at least one surfactant, approximately 0 or 0.05 to 10 g/l of anions of carbonate, nitrate and/or sulphate, and 0 or 0.001 to 2 g/l of carboxylate and/or sulphonate anions, wherein the content of molybdate and/or of P-containing oxy anions is in each case <0.1 g/l or is approximately 0 g/l, and wherein the composition has a pH value in the range from 2.5 to 6.5. The invention also relates to a corresponding coating and to the use of the substrates coated in this way.