Transient analysis has been investigated numerically to determine heat transfer by natural convection between concentric and vertically eccentric spheres with isothermal boundary conditions. The inner and outer spheres were heated and cooled in a step change of temperature. The governing equations, in terms of vorticity, stream function and temperature were expressed in a radial coordinate transformation coordinate system. The alternating direction implicit method and the successive over-relaxation techniques were applied to solve the finite difference form of governing equations. A physical model was introduced, which accounts for the effects of fluid buoyancy as well as eccentricity of the outer sphere. Transient solutions of the entire flow field were obtained for a range of Rayleigh numbers (10(3) < Ra < 10(5)), for a Prandtl number of 0.7 and a radius ratio of 2.0, with the outer sphere near the top and bottom of the inner sphere (epsilon = +/- 0.625). Results of the parametric study conducted further reveal that the heat and now fields are primarily dependent on the Rayleigh number and on the eccentricity of the annulus. Comparisons are attempted between the present computations and the results available in those of previous experimental and numerical studies.