A cosmetic composition for the care or make-up of the nails, comprising a functionalised polymer of the following formula ##STR27## in which the polymer chain (P) is a hydrophobic chain obtained by radical polymerisation of at least one ethylenically unsaturated monomer and whose number average molar mass is between 500 and 250,000 and wherein
- R represents a hydrogen atom or a linear or branched hydrocarbon chain having 1 to 8 carbon atoms optionally substituted with at least one group selected from CO.sub.2 H, NH.sub.2, OH or a phenyl group, itself being optionally substituted;
- A and B, identical or different, each represent a single bond, a saturated or unsaturated linear or branched hydrocarbon chain having from 1 to 16 carbon atoms, an amide bond or a peptide chain having 2 to 4 amino acids;
- the COOH and/or NH.sub.2 groups being free or salified.