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We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain Ω. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces K_μ,1^m(Ω). Using our a priori results, we give a specific construction of graded meshes on which the Taylor–Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.