A high-strength, cold-rolled steel sheet comprising: a chemical composition consisting of, by mass%, C: 0.060 % or more and 0.200 % or less, Si: 0.50 % or more and 2.20 % or less, Mn: 1.00 % or more and 3.00 % or less, P: 0.100 % or less, S: 0.0100 % or less, Al: 0.010 % or more and 2.500 % or less, N: 0.0100 % or less, Nb: 0.001 % or more and 0.200 % or less, V: 0.001 % or more and 0.200 % or less, and optionally one or more of Cr: 0.05 % or more and 1.00 % or less, Mo: 0.05 % or more and 1.00 % or less, Ni: 0.05 % or more and 1.00 % or less, Cu: 0.05 % or more and 1.00 % or less, B: 0.0003 % or more and 0.0050 % or less, Ca: 0.0010 % or more and 0.0050 % or less, Mg: 0.0005 % or more and 0.0100 % or less, REM: 0.0003 % or more and 0.0050 % or less, Sn: 0.0020 % or more and 0.2000 % or less, Sb: 0.0020 % or more and 0.2000 % or less and Ta: 0.0010 % or more and 0.1000 % or less, and the balance consisting of Fe and incidental impurities, the chemical composition satisfying a relation of 500 ≤ C* ≤ 1300, where C* is determined by formula given below; and a microstructure that contains ferrite in an area ratio of 20 % or more, martensite in an area ratio of 5 % or more, and tempered martensite in an area ratio of 5 % or more, the ferrite having a mean grain size of 20.0 μm or less, an inverse intensity ratio of γ-fiber to α-fiber in the ferrite being 1.00 or more and an inverse intensity ratio of γ-fiber to α-fiber in the martensite and the tempered martensite being 1.00 or more, and the steel sheet has a tensile strength of 780 MPa or more, wherein the area ratio of ferrite, martensite and tempered martensite; the inverse intensity ratio of γ-fiber to α-fiber in the ferrite, in the martensite and in the tempered martensite; and the tensile strength are measured in accordance with the description: C * = C − 12.0 / 92.9 × Nb − 12.0 / 50.9 × V − 12.0 / 180.9 × Ta × 10000 where each of the element symbols C, Nb, V, and Ta indicates content by mass% of the corresponding element in the steel sheet, and the unit of C* is mass ppm. a chemical composition consisting of, by mass%, C: 0.060 % or more and 0.200 % or less, Si: 0.50 % or more and 2.20 % or less, Mn: 1.00 % or more and 3.00 % or less, P: 0.100 % or less, S: 0.0100 % or less, Al: 0.010 % or more and 2.500 % or less, N: 0.0100 % or less, Nb: 0.001 % or more and 0.200 % or less, V: 0.001 % or more and 0.200 % or less, and optionally one or more of Cr: 0.05 % or more and 1.00 % or less, Mo: 0.05 % or more and 1.00 % or less, Ni: 0.05 % or more and 1.00 % or less, Cu: 0.05 % or more and 1.00 % or less, B: 0.0003 % or more and 0.0050 % or less, Ca: 0.0010 % or more and 0.0050 % or less, Mg: 0.0005 % or more and 0.0100 % or less, REM: 0.0003 % or more and 0.0050 % or less, Sn: 0.0020 % or more and 0.2000 % or less, Sb: 0.0020 % or more and 0.2000 % or less and Ta: 0.0010 % or more and 0.1000 % or less, and the balance consisting of Fe and incidental impurities, the chemical composition satisfying a relation of 500 ≤ C* ≤ 1300, where C* is determined by formula given below; and a microstructure that contains ferrite in an area ratio of 20 % or more, martensite in an area ratio of 5 % or more, and tempered martensite in an area ratio of 5 % or more, the ferrite having a mean grain size of 20.0 μm or less, an inverse intensity ratio of γ-fiber to α-fiber in the ferrite being 1.00 or more and an inverse intensity ratio of γ-fiber to α-fiber in the martensite and the tempered martensite being 1.00 or more, and the steel sheet has a tensile strength of 780 MPa or more, wherein the area ratio of ferrite, martensite and tempered martensite; the inverse intensity ratio of γ-fiber to α-fiber in the ferrite, in the martensite and in the tempered martensite; and the tensile strength are measured in accordance with the description: C * = C − 12.0 / 92.9 × Nb − 12.0 / 50.9 × V − 12.0 / 180.9 × Ta × 10000 where each of the element symbols C, Nb, V, and Ta indicates content by mass% of the corresponding element in the steel sheet, and the unit of C* is mass ppm.