In this thesis, the plasmon hybridization method is extended theoretically to explore the optical properties of curvilinear particles, high symmetry clusters, and infinite periodic systems of nanoparticles. Plasmon hybridization is a recently-developed theory used to describe the collective oscillations of conduction electrons in metallic nanoparticles (plasmons). Here curvilinear particles refer to solid nanoparticles, dielectric cavities in an infinite metal, or nanoparticles consisting of a dielectric core surrounded by a thin metallic shell (core/shell particles) that can be described using a coordinate system with a completely separable solution to the Laplace equation. I find that there is a common form for the plasmon frequencies of such particles among all completely separable coordinate systems and that the plasmons of core/shell particles can be viewed as a hybridization resulting from the interaction of solid particle and cavity plasmons. I specifically analyze the plasmons of prolate, oblate, and cylindrical particles, three experimentally relevant geometries. High symmetry clusters are collections of nanoparticles that exhibit the symmetry of a point group. I study the plasmons of nanosphere trimers (equilateral triangles, group D3 h), quadrumers (squares, group D4 h), and tetramers (tetrahedra, group Td). This study shows that the plasmons of these systems are composed of linear combinations of plasmons from each individual particle and may be classified into the irreducible representations corresponding to the point group to which each system belongs. This represents a step forward in understanding the underlying concepts behind the plasmon modes of multi-particle systems. The periodic systems that are examined in this thesis include a one-dimensional infinite nanosphere chain and two-dimensional hexagonal and square nanosphere arrays. The calculated plasmon energies are shown to agree very well with Finite Difference Time Domain calculations, a somewhat surprising result considering the quasistatic nature of plasmon hybridization. In contrast to other modeling methods, wherein a nanosystem's plasmon frequencies are calculated computationally or as the poles of a polarizability function, plasmon hybridization provides a physical picture of the plasmon modes in each system in analogy with molecular orbital theory and thus proves to be an essential tool in understanding the fundamental science behind the plasmonics of these systems.