A propylene-based polymer composition comprising: 10-95 parts by weight of a syndiotactic propylene polymer, and 90 to 5 parts by weight of a propylene α-olefin copolymer (provided that the sum of amounts of and is 100 parts by weight), wherein the polymer satisfies the following requirement (a) and the copolymer satisfies the following requirement (b): (a) the syndiotactic pentad fraction (rrrr fraction) as measured by 13 C-NMR is 85% or greater, the melting point (Tm) as determined by DSC is 145°C or higher, and constituent units derived from propylene are contained in an amount of more than 90 mol% (provided that the total amount of the constituent unit in the polymer is 100 mol%), and (b) constituent units derived from propylene are contained in an amount of 30 to 90 mol%, constituent units derived from at least one olefin selected from α-olefins having 2 to 20 carbon atoms (excluding propylene) are contained in an amount of 10 to 70 mol% (provided that the sum of amounts of the constituent units derived from propylene and the constituent units derived from the α-olefin having 2 to 20 carbon atoms (excluding propylene) is 100 mol%), the MFR measured according to JIS K-6721 at 230°C under a load of 2.16 kg is in the range of 0.01 to 100 g/10 min, and at least one of the following requirements (b-1) and (b-2) is satisfied: (b-1) the syndiotactic triad fraction (rr fraction) as measured by 13 C-NMR is 60% or greater, and (b-2) the intrinsic viscosity [η] (dL/g) as measured in decalin at 135°C and the MFR (g/10 min, 230°C, under a load of 2.16 kg) satisfy the following relationship: 1.50 × MFR − 0.20 ≤ η ≤ 2.65 × MFR − 0.20. 10-95 parts by weight of a syndiotactic propylene polymer, and 90 to 5 parts by weight of a propylene α-olefin copolymer (provided that the sum of amounts of and is 100 parts by weight), (a) the syndiotactic pentad fraction (rrrr fraction) as measured by 13 C-NMR is 85% or greater, the melting point (Tm) as determined by DSC is 145°C or higher, and constituent units derived from propylene are contained in an amount of more than 90 mol% (provided that the total amount of the constituent unit in the polymer is 100 mol%), and (b) constituent units derived from propylene are contained in an amount of 30 to 90 mol%, constituent units derived from at least one olefin selected from α-olefins having 2 to 20 carbon atoms (excluding propylene) are contained in an amount of 10 to 70 mol% (provided that the sum of amounts of the constituent units derived from propylene and the constituent units derived from the α-olefin having 2 to 20 carbon atoms (excluding propylene) is 100 mol%), the MFR measured according to JIS K-6721 at 230°C under a load of 2.16 kg is in the range of 0.01 to 100 g/10 min, and at least one of the following requirements (b-1) and (b-2) is satisfied: (b-1) the syndiotactic triad fraction (rr fraction) as measured by 13 C-NMR is 60% or greater, and (b-2) the intrinsic viscosity [η] (dL/g) as measured in decalin at 135°C and the MFR (g/10 min, 230°C, under a load of 2.16 kg) satisfy the following relationship: 1.50 × MFR − 0.20 ≤ η ≤ 2.65 × MFR − 0.20. (b-1) the syndiotactic triad fraction (rr fraction) as measured by 13 C-NMR is 60% or greater, and (b-2) the intrinsic viscosity [η] (dL/g) as measured in decalin at 135°C and the MFR (g/10 min, 230°C, under a load of 2.16 kg) satisfy the following relationship: 1.50 × MFR − 0.20 ≤ η ≤ 2.65 × MFR − 0.20.