The adhesive ethylene copolymer composition of the invention comprises a specific long-chain branched ethylene/&agr;-olefin copolymer (or this copolymer and its graft-modified product) and a tackifier (or a specific ethylene/vinyl acetate copolymer), and has specific density, MFR, crystallinity and graft quantity. The composition containing the tackifier can form a layer having good adhesive force to polystyrene resins, ABS resin, polyacrylonitrile resins and ethylene/vinyl acetate copolymer saponified resins even in an atmosphere of high temperatures. The composition containing the ethylene/vinyl acetate copolymer can form a layer having good adhesive force to polyester resins, polycarbonate resins, polyvinylidene chloride resins and ethylene/vinyl acetate saponified resins even in an atmosphere of high temperatures. The other adhesive ethylene copolymer composition of the invention comprises a blend of a graft-modified linear ethylene/&agr;-olefin copolymer and an olefin elastomer. This composition has excellent heat stability, moldability and transparency, and can form a film layer having good adhesive force to metals and materials of high polarity.