A chlorosulphonated ethylene-.alpha.-olefin copolymer having a chlorine content of 0.5-20% by weight and a glass transition temperature in differential scanning calorimetry (DSC) of -50.degree. C. or lower and being crystalline. The chlorosulphonated ethylene-.alpha.-olefin copolymer is obtained by chlorinating a crystalline ethylene-.alpha.-olefin copolymer having a density of 890 kg/m.sup.3 or lower, obtained by copolymerizing .alpha.-olefin comprising a hydrocarbon having 4 or more carbon atoms, with ethylene, and then chlorosulphonating the chlorinated copolymer.