To understand fullerenes, it is crucial to determine the energies, and hence the relative stabilities, of different structures. Here we introduce an approach to calculating and understanding these energies, by applying elasticity theory. Explicit formulas are given for the energies of ball-shaped molecules, graphite tubules, and the recently proposed negative-curvature fullerenes. Extensive atomistic calculations confirm the validity of the formulas. These results provide a general perspective on the relative stability of the different fullerenes