The propylene polymer of the present invention satisfies (1) a 25.degree. C. hexane soluble content (H25) of 0-80 wt %; and (2) either no melting temperature (Tm) measurable by differential scanning calorimetry (DSC), or a melting temperature (Tm) satisfying, if measurable by DSC, the following relationship: .DELTA.H.gtoreq.3.times.(Tm-120) wherein .DELTA.H is a melting endotherm (J/g). The propylene homopolymer of the present invention satisfies (1) a meso pentad fraction (mmmm) of 30-60 mol %; (2) a racemic pentad fraction (rrrr) satisfying the following relationship: [rrrr/(1-mmmm)].ltoreq.0.1; (3) a fraction (W25) eluted at a temperatures up to 25.degree. C. by temperature-programmed chromatography, of from 20-100 wt %; and, (4) a pentad fraction (rmrm) of more than 2.5 mol %. The propylene copolymer of the present invention satisfies (1) a stereoregularity index (P) of 55-90 mol % as determined by .sup.13C-NMR measurement; and (2) a fraction (W25) eluted at a temperatures up to 25.degree. C. by temperature-programmed chromatography, of from 20-100 wt %.