  We consider the Casimir interaction between two spheres at zero and finite
temperature, for both scalar fields and electromagnetic fields. Of particular
interest is the asymptotic expansions of the Casimir free energy when the
distance between the spheres is small. The scenario where one sphere is inside
the other is discussed in detail. At zero temperature, we compute analytically
the leading and the next-to-leading order terms from the functional determinant
representation of the Casimir energy. As expected, the leading order term
agrees with the proximity force approximation. The results for the
next-to-leading order terms are new. In the limit where the radius of the outer
sphere goes to infinity, the results for the sphere-plane geometry are
reproduced. At finite temperature, the leading order term is computed and it is
found to agree completely with the proximity force approximation in the medium
and high temperature regions. For the scenario where two spheres are outside
each other, analogous results are obtained. In the case of Dirichlet boundary
conditions on both spheres, the next-to-leading order term of the zero
temperature Casimir energy is found to agree with that computed recently using
derivative expansion.
