This article provides a review of our recent Rayleigh scattering measurements on single metal nanoparticles. Two different systems will be discussed in detail: gold nanorods with lengths between 30 and 80 nm, and widths between 8 and 30 nm; and hollow gold-silver nanocubes (termed nanoboxes or nanocages depending on their exact morphology) with edge lengths between 100 and 160 nm, and wall thicknesses of the order of 10 nm. The goal of this work is to understand how the linewidth of the localized surface plasmon resonance depends on the size, shape, and environment of the nanoparticles. Specifically, the relative contributions from bulk dephasing, electron-surface scattering, and radiation damping (energy loss via coupling to the radiation field) have been determined by examining particles with different dimensions. This separation is possible because the magnitude of the radiation damping effect is proportional to the particle volume, whereas, the electron-surface scattering contribution is inversely proportional to the dimensions. For the nanorods, radiation damping is the dominant effect for thick rods (widths greater than 20 nm), while electron-surface scattering is dominant for thin rods (widths less than 10 nm). Rods with widths in between these limits have narrow resonances-approaching the value determined by the bulk contribution. For nanoboxes and nanocages, both radiation damping and electron-surface scattering are significant at all sizes. This is because these materials have thin walls, but large edge lengths and, therefore, relatively large volumes. The effect of the environment on the localized surface plasmon resonance has also been studied for nanoboxes. Increasing the dielectric constant of the surroundings causes a red-shift and an increase in the linewidth of the plasmon band. The increase in linewidth is attributed to enhanced radiation damping.