The interest in and study of compact, co-dimension one minimizers has at least a century-old history: in 1904, J. J. Thomson proposed minimizing the electrostatic potential over sets of particles restricted to a sphere as part of his model of the atom. Modern physical examples of these assemblies occur in the realm of interacting nanoparticles. Many species of virus rely on the formation of a hollow sphere to enclose and deliver their genetic material, for example. Inorganic polyoxometalate (POM) macroions also form into hollow spherical structures in a similar way. I will discuss recently developed mathematical theory that characterizes when spherical assemblies define energy favorable structures, as well as applications to physical models of these assemblies.