The invention relates to a process for coating metallic surfaces with an aqueous composition as solution or as dispersion, wherein the composition contains at least one phosphate, at least 3 g/l of at least one titanium or/and zirconium compound and at least one complexing agent, and also corresponding aqueous compositions. The coatings produced therewith display very good bare corrosion protection in the NSS salt spray test and in the condensate water constant temperature and humidity test.