A propylene polymer is characterized by satisfying the following (1) to (5): (1) a melt flow rate (MFR) measured at 230° C. under a 2.16 kg load of from 0.1 to 1000 g/10 min; (2) an isotactic triad traction (mm) measured by 13C-NMR of 99.0 % or above; (3) a Q value measured by gel permeation chromatography (GPC) of from 2.0 to 6.0; (4) a relation between melt flow rate (MFR: g/10 min) measured at 230° C. under a 2.16 kg load and a memory effect (ME) measured at 190° C. at an orifice diameter of 1.0 mm satisfying the following formula (I): 1.75≧(ME)+0.26×log(MFR)≧1.40  (I); and (5) a relation of the cold xylene solubles at 23° C. (CXS) satisfying the following formula (II) CXS≦0.5×[C2]+0.2×log(MFR)+0.5  (II) wherein [C2] represents the ethylene unit content (% by weight) in the polymer.