The propylene polymer of the invention is transparent and has good low-temperature heat-sealability and scratch resistance, and is favorable to wrapping and packaging films. (1) The polymer has a melting point, Tm (° C.), measured through differential scanning calorimetry, of 110≦Tm≦140; (2) its heat of fusion &Dgr;H (J/g) and melting point Tm (° C.) satisfy &Dgr;H≧0.45×Tm+22; (3) the half-value width Th (° C.) of the peak top of its elution curve obtained in programmed-temperature fractionation is Th≦5; and (4) its intrinsic viscosity [&eegr;] (dl/g) measured in a solvent of tetralin at 135° C. falls between 0.5 and 5.