Periodic arrays of proximate but noncontacting noble metal nanoparticles provide the ability to electromagnetically couple the surface plasmons on various nanoparticles to produce collective delocalized surface-plasmon-polariton modes with well characterized mode structure. Such structures provide intrinsic interest as well as provides an attractive concept for optical sensing in potentially open structures permitting gas or liquid to percolate through the structure. In this talk, I review our work on modeling the surface-plasmon-polariton modes of various nanoparticle arrays. I discuss one-dimensional chains and two-dimensional nanoparticle slabs. Ways of dealing with losses will be discussed. Provided the ratio of nanoparticle spacing to nanoparticle diameter (for spherical nanoparticles) exceeds a critical value, the electromagnetic interactions between nanoparticles are dominated by dynamic dipole-dipole coupling. In other words, one nanoparticle acts on the others (and on itself) by means of the retarded electromagnetic dipole-dipole interactions between all nanoparticles. The resulting collective modes delocalized over the structure are surface-plasmon polaritons. We consider various structures including infinite and finite chains, ring-shaped arrangements of nanoparticles, and two-dimensional slab arrays. Given the dispersion of surface-plasmon polaritons, we can then obtain the linear optical properties (e.g., reflection and transmission spectra, angle-dependence of reflectivity and transmittivity), which in turn provide important information on the design of nanoparticle-array-based devices. Surface-plasmon polaritons are strongly attenuated by homogeneous and inhomogenous broadening as well as by intrinsic radiative loss. We discuss strategies for mitigating these losses.