Ethylene-a-olefin copolymer rubber comprising ethylene and a-olefin having 3 to 6 carbon atoms, or ethylene, a-olefin having 3 to 6 carbon atoms and a non-conjugated polyene, the molar ratio of ethylene/a-olefin being 40/60 to 95/5, the content of non-conjugated polyene being 0 to 25% by weight, the intrinsic viscosity as measured in xylene at 70°C being 0.8 to 5.0 dl/g, the Q value (weight average molecular weight/number average molecular weight) being 4 to 20, and the peak characteristic numbers I and A(1) obtained from molecular weight distribution curve being