An ethylene/.alpha.-olefin copolymer obtained by copolymerizing ethylene with an .alpha.-olefin of 3 to 20 carbon atoms in the presence of a catalyst for olefin polymerization comprising an organoaluminum oxy-compound (a) and at least two kinds of transition metal compounds (b). This ethylene/.alpha.-olefin copolymer has high melt tension and high flowability. A graft modified ethylene/.alpha.-olefin copolymer obtained by graft polymerizing the above-mentioned ethylene/.alpha.-olefin copolymer with a polar monomer. An ethylene copolymer composition comprising the above-mentioned graft modified ethylene/.alpha.-olefin copolymer and polyolefin. The ethylene/.alpha.-olefin copolymer of the invention is excellent in moldability. From this copolymer, a film having high transparency and high mechanical strength can be produced. The graft modified ethylene/.alpha.-olefin copolymer of the invention is excellent in heat stability and moldability. From this copolymer, a film having high transparency and good adhesion to highly polar materials can be produced. The ethylene copolymer composition of the invention is excellent in heat stability and moldability. From this composition, a film having high transparency and good adhesion to highly polar materials can be produced.