Estimation of the Rayleigh numbers of the inner-core leads to the conclusion that, even with low internal heating rates, this region of the Earth is convecting. This work presents an axisymmetrical model able to investigate convection in a sphere and its applications to the Earth's inner-core. Rayleigh numbers ranging from the critical (onset of the convection) to ten thousand times the critical have been explored. The thermal implications of a convecting inner-core have been pointed out: in all the cases the temperature profiles are very close to the adiabat, leading to the possibility of a partially molten regions in the inner-core. Indeed, the most important effects appear in the first tens of kilometers, just below the inner-core surface and could be at the origin of the low quality factors Q&agr; and Q&bgr; revealed by seismological studies. ¿ American Geophysical Union 1992