Many materials grown by Bridgman solidificatian have anisotropic solid-phase thermal conductivities. To date, computational simulation of directional solidification of these materials has accounted for conduction heat transfer in the melt and solid, and for phase change. We develop a computational model for steady axisymmetric Bridgman solidification of anisotropic materials that includes buoyancy-driven convection and shrinkage flow in the melt, and for conduction in the ampoule. The results are illustrated for benzene, for which the thermal conductivity has been measured parallel and perpendicular to the growth direction of directionally solidified specimens. We study the effects of anisotropy by performing computations using measured components of the thermal conductivity tensor, and by considering macroscopically isotropic (e.g., polycrystalline) solid benzene, as well as fictitious materials whose properties differ from those of benzene only in that they have a larger solid-phase conductivity in the direction of growth. Besides concave and convex interface shapes commonly associated with vertical Bridgman growth, we also find, for a range of solidification conditions, interface shapes that are neither convex nor concave. We find that increasing the conductivity parallel to the growth direction has the effect of planarizing the interface over a large central core, and increasing the curvature near the wall. The differences between our computations and those not accounting for buoyancy-driven and solidification shrinkage flow in Bridgman solidification of anisotropic materials an discussed.