A propylene copolymer composition enabling the production of a film having excellent low-temperature heat sealing ability and blocking resistance, is obtained by blending 1 to 70 wt % of a component A and 30 to 99 wt % of a component B. The component A comprises either a propylene homopolymer or a copolymer of propylene with not more than 10 mol % of ethylene or an a-olefin having a carbon number of 4 to 20, and has a molecular weight distribution (Mw/Mn) of not more than 3, a mean elution temperature of 40.degree. to 100.degree. C., and an elution dispersion degree (.sigma. value) of not more than 10. The component B comprises a copolymer of propylene with ethylene and/or an .alpha.-olefin having a carbon number of 4 to 20, and has a molecular weight distribution (Mw/Mn) of 3.5 to 10.