Abstract: A terpolymer rubber prepared from ethylene, .alpha.-olefin having 3 to 20 carbon atoms and branched chain polyene of the formula ##STR1## wherein R.sup.1 represents C.sub.1 -C.sub.5 alkyl, and R.sup.2 and R.sup.3 represent hydrogen atom or C.sub.1 -C.sub.5 alkyl, such as, for example, 6,7-dimethyl-4-ethylidene-1,6-octadiene, 6,7-dimethyl-4-ethylidene-1,6-nonadiene, and 9-methyl-4-ethylidene-1,8-decadiene; when added to a rubber (diene rubber, nitrile rubber or random copolymer rubber from ethylene, C.sub.3 -C.sub.20 .alpha.-olefin, and optionally a nonconjugated diene) provides curable rubber compositions having rapid vulcanization rates, for example, as lows as about 7 to 10 seconds. The diene rubber compositions also have excellent resistances to weather,ozone and heat aging while maintaining excellent mechanical strength characteristics. The nitrile rubber compositions also have excellent oil resistance.