The fundamental solutions for boundary element analysis of torsionai problems of a cylindrical anisotropic medium are presented. The particular form of cylindrical anisotropy considered in this paper is the most general kind possible for axisymmetric torsion deformation. A suitable change of variables is suggested to reduce the governing equation to a simple form which is amenable to Hankel transforms. The explicit form of the fundamental solution for infinite space subjected to an axisymmetric ring of uniformly applied tangential forces is then derived. An existing solution for an isotropic medium is recovered. In addition, simple half‐space fundamental solutions for the torsion of a transversely isotropic medium are determined. The kernel functions obtained herein are readily implemented in the boundary element formulation of the axisymmetric torsionai problems.