A theory of the coupling of evanescent optical fields between metallic nanoparticles is developed to provide a basis for designing plasmonic systems. The interaction between metallic nanoparticles is investigated using an electrostatic approximation that describes the localized surface-plasmon resonances for particles much smaller than the wavelength of the exciting radiation. A coupling coefficient is derived relating the surface charge on one particle to the induced surface dipoles on another. The effect of dipole radiation damping is included to model the radiation broadening of the plasmon resonances. The theory enables an analysis of the key factors that control the coupling and that lead to resonances of the particle system. It is shown that the coupling between a simple pair of particles, in the form of stripes, leads to frequency splitting and the formation of a dark mode. The dark mode is associated with little scattering of light but large evanescent electric fields. It is shown that the dark mode has low radiation damping compared to an associated bright mode. The theory can be applied to an arbitrary number of interacting particles, allowing them to be configured to achieve the desired properties of the plasmonic system.