The present invention describes novel statistical copolymers and the preparation and possible uses thereof. Copolymers are described that are water-soluble and cross-linkable and that comprise the copolymerization product of a monomer mixture consisting substantially of a vinyl lactam (a) and at least one further vinyl monomer (b) of a different type selected from the group consisting of hydrophobic, hydrophilic and functional vinyl monomers, wherein the monomers are present in the copolymer in the form of statistically distributed building blocks, and, if a functional vinyl monomer is present as a building block in the copolymer, that building block is, where appropriate, modified with a reactive vinyl monomer (c), the reactive vinyl monomer (c) being linked to a building block of a functional vinyl monomer with retention of its vinylic group and with the formation of a covalent bond.