A mechanism leading to the spin-triplet superconductivity is proposed based on the antiferromagnetic spin fluctuation. The effects of anisotropy in spin fluctuation on the Cooper pairing and on the direction of d vector are examined in the one-band Hubbard model with RPA approximation. The gap equations for the anisotropic case are derived and applied to Sr2RuO4. It is found that a nesting property of the Fermi surface together with the anisotropy leads to the triplet superconductivity with the d =ẑ(sin kx ± i sin ky), which is consistent with experiments.74. 74.20Mn, 74.25Dw Since the discovery of superconducting phase in Sr 2 RuO 4 [1], much effort has been paid for understanding its exotic properties. Among several interesting natures, the most fascinating one is that it is a spintriplet superconductor confirmed by NMR experiment [2]. While most superconductors found during several decades are singlet, the only exceptions were 3 He and UPt 3 . Therefore the fact that the triplet pairing is realized in Sr 2 RuO 4 has attracted much attention. While UPt 3 , the second example of spin-triplet superconductor, has a complicated electronic structure, Sr 2 RuO 4 has a rather simple electronic state [1]. Thus clarifying the microscopic mechanism of superconductivity in Sr 2 RuO 4 is very important for understanding the triplet superconductors in general.In 3 He, Cooper pairs are formed due to ferromagnetic spin fluctuations peaked at q = 0 [3,4]. Therefore it is natural to expect the origin of the triplet pairing in Sr 2 RuO 4 is also ferromagnetic spin fluctuation [5,6]. This assumption has been believed to be justified by NMR experiments [7][8][9]. However the recent neutron scattering experiment has shown that there exists a significant peak near q 0 = (±2π/3, ±2π/3) and no sizable ferromagnetic spin fluctuation [10]. Thus it is difficult to assume that the spin fluctuation near q 0 plays no role in the Cooper pairing in Sr 2 RuO 4 . (In the following discussion we call this fluctuation as antiferromagnetic (AF) spin fluctuation, for simplicity.) However this AF fluctuation leads to the singlet superconductivity rather than * e-mail address: t-kuwabara@bea.hi-ho.ne.jp † Address from April 2000, Dept. of Physics, Univ. of Tokyo, Hongo, Bunkyo-ku 113-0033 Tokyo the triplet superconductivity as expected in analogy to high-T c cuprates [5].In this paper we propose a mechanism which gives the triplet pairing even if the spin fluctuation is AF. We find that the characteristic features of Sr 2 RuO 4 are twofold: One is the anisotropy of the spin fluctuation found in NMR experiments [8,9], and the other is a nesting property with momentum q 0 of the two-dimensional Fermi surface. We show that these two features explain the pairing in Sr 2 RuO 4 .In addition to the competition between singlet and triplet pairing, the direction of the d vector, which is the order parameter of triplet superconductivity, is another interesting problem. We show that the anisotropy of the spin fluctuation also explains the experimental fact...