Radial curvature effects on axisymmetric free convection boundary-layer flow are investigated for vertical cylinders and cones for some special non-uniform temperature differences between the surface and the ambient fluid. The solution is given as a power series expansion, the first term being equal to the solution to be found when no transverse curvature is involved. For a variety of Prandtl numbers numerical integrations have been carried out. An interesting application is the determination of the surface temperature of a free-convection-cooled cylinder with constant heat flux through the surface. For the Prandtl numbers considered, it turns out that the same heat flux induces a lower surface temperature on a cylinder than on a flat plate.