We study anisotropic cylindrical dielectric waveguides under the following conditions: (1) their cross section Ω has arbitrary shape; (2) the dielectric permittivity tensor ε has discontinuities at the boundaries, as well as a graded spatial variation; (3) Ω is small, and ε minus the unit 3 × 3 matrix (I) is small and smooth, so that only one pair of propagation modes is allowed. A general and exact dispersion relation for the propagation modes is derived, using integral equation techniques and the long wavelength singularity of the free-space Green's function in two dimensions. We present approximate explicit solutions of the dispersion relation to first order in (ε−I), which exhibit a logarithmic dependence and birefringence, in general. Some numerical estimates are also given. Approximate dispersion relations up to and including second order in (ε −I) are also discussed. The scattering problem is treated briefly.