Disclosed is a process for preparing an ethylene-.alpha.-olefin-nonconjugated polyene random copolymer comprising random copolymerizing (a) ethylene, (b) an .alpha.-olefin of 3 or more carbon atoms and (c) a nonconjugated polyene in the presence of a metallocene catalyst containing a specific metallocene compound. The ethylene-.alpha.-olefin-nonconjugated polyene random copolymer prepared by the process has such properties that (i) the copolymer contains units derived from ethylene (a) and units derived from the .alpha.-olefin of 3 or more carbon atoms (b) in a molar ratio of 40/60 to 95/5 �(a)/(b)!, (ii) the iodine value is 1 to 50, (iii) the intrinsic viscosity (.eta.) is more than 0.1 dl/g and less than 8.0 10 dl/g, (iv) the intensity ratio D of T.alpha..beta. to T.alpha..alpha. in the .sup.13 C-NMR spectrum, T.alpha..beta./T.alpha..alpha., is not more than 0.5, (v) the B value is 1.00 to 1.50, (vi) the glass transition temperature Tg is not higher than -50.degree. C., and (vii) the ratio g.eta.* of the intrinsic viscosity (.eta.) of this copolymer defined above to the intrinsic viscosity (.eta.).sub.blank of a straight-chain ethylene-propylene copolymer having the same weight-average molecular weight as this random copolymer and having an ethylene content of 70% by mol, (.eta.)/(.eta.).sub.blank, is more than 0.9. Also disclosed is a vulcanizable rubber composition containing the random copolymer. The ethylene-.alpha.-olefin-nonconjugated polyene random copolymer and the rubber composition are excellent in mechanical strength, weathering resistance, ozone resistance, processability, low-temperature flexibility and heat resistance.