The present invention relates to a chromium-free aqueous agent based on water-soluble compounds of titanium and/or zirconium and a source of fluoride ions, copper ions and metal ions selected from the group consisting of calcium, magnesium, aluminum, boron, zinc, iron, manganese and/or tungstene and to a method for the anti-corrosive conversion treatment of metal surfaces. The chromium-free aqueous agent is suitable for the treatment of various metal materials, joined in composite structures, amongst others of steel or galvanized steel or the alloys thereof or any combinations of said materials. Furthermore, surfaces of aluminum and alloys thereof can be treated in an anti-corrosive manner using the agent according to the invention. The anti-corrosive treatment is intended in particular as a pretreatment for a subsequent dip-coating. The invention further relates to a metallic substrate that was treated according to a predefined method sequence with the chromium-free agent according to the invention and to the use thereof, particularly in the automotive production of vehicle bodies.