The random, hydrogenated copolymers according to the present invention of unsubstituted conjugated dienes and substituted conjugated dienes, having a glass transition temperature of ≦−60° C., an enthalpy of fusion of ≦100 J/g and a content of residual double bonds in the range from 0 to 80%, are suitable for the production of rubber moldings of all kinds which have good resistance to ageing and high resilience at low temperatures. In particular, various tire components and industrial rubber articles may be mentioned as rubber moldings. The copolymers are also suitable for the rubber modification of brittle thermoplastics.