We consider the vorticity-stream formulation of axisymmetric incompressible flows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector field in terms of the swirling components, an extra set of pole conditions is necessary to give a full description of the regu larity. In addition, smooth solutions up to the axis of rotation give rise to smooth solutions of primitive formulation in the case of the Navier-Stokes equation, but not the Euler equation. We also establish a proper weak formulation and show its equivalence to Leray's formulation. © 2009 Society for Industrial and Applied Mathematics.