The Casimir force for charge-neutral, perfect conductors of non-planar geometric configurations have been investigated. The configurations were: (1) the plate-hemisphere, (2) the hemisphere-hemisphere and (3) the spherical shell. The resulting Casimir forces for these physical arrangements have been found to be attractive. The repulsive Casimir force found by Boyer for a spherical shell is a special case requiring stringent material property of the sphere, as well as the specific boundary conditions for the wave modes inside and outside of the sphere. The necessary criteria in detecting Boyer's repulsive Casimir force for a sphere are discussed at the end of this thesis.