Abstract: The invention relates to the selective hydrogenation of unsaturated polymers, in particular elastomeric polymers which contain aromatic rings and carbon-carbon double rings, for example polymers prepared from styrene and butadiene. In this case, the polymer molecule contains internal and terminal-group double bonds, as well as unsaturated bonds in aromatic rings. According to the invention, only the unsaturated terminal and internal bonds are hydrogenated, whereby the thermal and weather resistance properties of the product are improved. Conventionally, hydrogenation has been performed by means of nickel or cobalt catalysts, but now it has been observed that a metallocene/alumoxane catalyst complex not only catalyzes the polymerization reaction of various olefinic compounds but also catalyzes the selective hydrogenation of unsaturated polymers, preferably styrene-butadiene-styrene block polymers.