Existence and uniqueness of global regular special solutions to Navier-Stokes equations with boundary slip conditions in axially symmetric domains is proved. The proof of global existence relies on the global existence results for axially symmetric solutions which were obtained by Ladyzhenskaya and Yudovich-Ukhovskiĭ in 1968 who employed the problem for vorticity. In this paper the equations for vorticity also play a crucial role. Moreover, the boundary slip conditions imply appropriate boundary conditions for vorticity, which is of crucial importance to this paper. Finally, we prove the existence of solutions which remain close to the axially symmetric solutions for all time. The existence of axially symmetric solutions was proved in weighted Sobolev spaces with the weight equal to a power of the distance to the axis of symmetry because in such spaces a global estimate for vorticity could be obtained. Therefore in this paper similar weighted Sobolev spaces are also used.