The paper presents theoretical and numerical investigations on simulation methods for the Boltzmann equation with axisymmetric geometry. The main task is to reduce the computational effort by taking advantage of the symmetry in the solution of the Boltzmann equation.; The reduction automatically leads to the concept of weighting functions for the radial space coordinate and therefore to a modified Boltzmann equation. Consequently the classical simulation methods have to be modified according to the new equation.; The numerical results shown in this paper - rarefied gas flows around a body with axisymmetric geometry - were done in the framework of the European space project HERMES.