Surface plasmon resonance is a kind of electromagnetic resonances that exists when there is an interface between metal and dielectric. Such kind of resonances is associated with the collective motion of electrons in metal. For an isolated metal nanoparticle, plasmons are localized on its surface. When metal nanoparticles are close enough, the plasmonic modes on one particle can couple with that on other particles so that hopping of surface plasmons from one particle to other particles becomes possible. When metal nanoparticles are arranged in a periodic array, there exist plasmonic bands at frequencies (from near inferred to ultraviolet range) that allow plasmons to propagate through the array. One important issue in both theoretical and experimental aspects is that there exist significant absorption losses in metallic nanostructures near optical frequencies. In addition, radiation loss can be significant in one- and two-dimensional arrays. This thesis is devoted to the development of the theory of resonance that can be applied efficiently to understand and predict the plasmonic modes of metal nanoparticle arrays. A dipolar eigen-decomposition theory is employed in this thesis and is further developed to introduce new concepts of normal modes and band structures in lossy systems. The plasmonic modes in single periodic chain, a pair of chains, and circular arrays are studied using the eigen-decomposition theory and the results are verified by numerical solutions calculated using rigorous multiple scattering theory. Results of metal nanoparticle chains are also compared to higher dimensional arrays. Finally, the effect of plasmonic resonance in a three dimensional chiral array is investigated.