Surface plasmon resonances in nanoparticles have numerous promising scientific and technological applications in such areas as nanophotonics, near-field microscopy, nano-lithography, biosensor, metamaterial and optical data storage. Consequently, the understanding of plasmon resonances in nanoparticles has both fundamental and practical significance. In this dissertation, a new numerical technique to fully characterize the plasmon resonances in three-dimensional nanoparticles is presented. In this technique, the problem of determining the plasmon resonant frequencies is framed as an integral equation based eigenvalue problem, and the plasmon resonant frequencies can be directly found through the solution of this eigenvalue problem. For this reason, it is computationally more efficient than other "trial-and-error" numerical techniques such as the finite-difference time-domain (FDTD) method. This boundary integral equation method leads to fully populated discretized matrix equations that are computationally expensive to solve, especially when a large number of particles are involved in the nanostructures. Since the fully populated matrices are generated by integrals with 1/r-type kernel, this computational problem is appreciably alleviated by using the fast multipole method (FMM). The boundary integral approach is also extended to the calculation of the extinction cross sections of nanoparticles, which reveal important information such as the strength and the full width at half maximum (FWHM) of these resonances. The numerical implementation of this technique is discussed in detail and numerous computational results are presented and compared with available theoretical and experimental data. Furthermore, metallic nanoshells for biosensing applications as well as nanoparticle-structured plasmonic waveguides of light are numerically investigated. The integral equation based numerical technique presented throughout this dissertation can be instrumental for the design of plasmon resonant nanoparticles and to tailor their optical properties for various applications.