A polymer suitable for use as a viscosity index improver for lubricating oil compositions, which is at least one, at least partially hydrogenated linear or star-shaped random copolymer of isoprene and butadiene, in which at least 70 wt. % of the butadiene is incorporated into the polymer as 1,4 units and the weight ratio of isoprene addition product to butadiene addition product in the copolymer is in a range of from about 90:10 to about 70:30.