The present invention relates to a composition for coating metallic surfaces, comprising a binder component; a corrosion inhibitor polymer based on an ethylenically monocarboxylic and dicarboxylic acid and, if appropriate, further ethylenical monomers, a solvent component, and, if appropriate, crosslinkable components, and also pigment/filler components. The invention further relates to processes for its preparation and also to methods of coating a metallic surface using the composition, to coated metallic surfaces thus obtained, and to the use of the composition as a primer, particularly in coil coating or in atmospheric corrosion control. After drying, the applied layer preferably has a thickness of at least 3.1 μm and is therefore thicker than a typical pretreatment layer.