Two radial–axial transient heat flow problems have been solved for regions bounded internally by circular cylinders. They are not of the simple "product-solution" type and it is considered that they may have application in other fields of physics where the Diffusion Equation applies. The problems arose during investigation into "end-effect" in cylindrical thermal-conductivity probes. The solutions are obtained by integral-transform methods, two different types of transform being used in each solution.