Disclosed is a polymer composition comprising a copolymer of propylene, a specific amount of .alpha.-olefin having 4 or more carbon atoms and optionally a specific amount of ethylene and a specific 1-butene homopolymer or copolymer having a melting point of not lower than 60.degree. C. but lower than 125.degree. C., wherein the polymer composition contains a 20.degree. C. xylene-soluble portion of the polymer composition in an amount of from 5 to 45% by weight and the 20.degree. C. xylene-soluble portion has an intrinsic viscosity of 1.3 dl/g or higher. The polymer composition may optionally contain a specific propylene copolymer and/or a specific propylene homopolymer or copolymer having a melting point of from 150.degree. C. to 170.degree. C.