Polymers Ia and Ib carrying functional groups and obtainable byA)1. hydrogenation of the olefinic double bonds of a polymer IIa comprisinga) from 20 to 100% by weight of a conjugated diene,b) from 0 to 80% by weight of an aromatic vinyl compound, andc) from 0 to 20% by weight of further comonomers,to give the hydrogenated polymer IIb,2. metallation of IIb using an organoalkali metal compound to give IIc,3. reaction of IIc with a diaziridine III to give IId, and4. hydrolysis or alcoholysis of IId to give Ia, or byB)1. partial hydrogenation of IIa to give a partially hydrogenated polymer IIb',2. epoxidation of the double bonds of IIb' to give IIe, and3. reaction of IIe with a compound IV carrying functional groups, to give Ib,are suitable as viscosity index improvers for engine oils.