We review the interaction of laser light with a single plasmonic nanostructure. Until recently, the extremely weak optical response of a single nanostructure had only allowed spectroscopic research on ensembles of nanoparticles, where both structural and material parameters vary from one nanoparticle to another and the optical response is therefore averaged over the ensemble. Measurements at the level of a single nanostructure provide an effective tool for investigating the fundamental optical and spectroscopic properties of nanostructures, allowing insight into the mechanisms of elementary physical processes and permitting one to avoid the averaging procedure with the inevitable loss of physical information. The diverse range of linear-interaction optical phenomena occurring in a weak light field is reviewed and nonlinear interaction with high-intensity radiation is discussed.