In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair ( ω r r , ω θ r ), we get several Prodi-Serrin type regularity criteria based on the angular velocity, u θ . Moreover, we obtain the global well-posedness result if the initial angular velocity u θ 0 is appropriate small in the critical space L 3 (R 3 ). Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say ω 3 or u 3 .