In this paper we overview our recent studies of anisotropic noble metal (e.g. gold and silver) nanoparticles, in which a combination of theory and experiment has been used to elucidate the extinction spectra of the particles, as well as information related to their surface enhanced Raman spectroscopy. We used wet-chemical methods to generate several structurally well-defined nanostructures other than solid spheres, including silver nanodisks and triangular nanoprisms, and gold nanoshells and multipods. When solid spheres are transformed into one of these shapes, the surface plasmon resonances in these particles are strongly affected, typically red-shifting and even splitting into distinctive dipole and quadrupole plasmon modes. In parallel, we have developed computational electrodynamics methods based on the discrete dipole approximation (DDA) method to determine the origins of these intriguing optical features. This has resulted in considerable insight concerning the variation of plasmon wavelength with nanoparticle size, shape and dielectric environment, as well as the use of these particles for optical sensing applications.