Some of the optical effects produced by plasmonic nanoparticles have been known since antiquity. However, the physical understanding of these properties only began in the first decade of the 20th century. The discovery of the dispersion relation of surface plasmons, almost 40 years later, unified the physical description of the plasmonic phenomena. Later, advances in nanofabrication initiated a revolution in the preparation of metallic nanostructures supporting surface plasmons. In parallel, modern electromagnetic computational methods have permitted the theoretical investigation of the optics of nanoscale and tailor plasmonic particles for experiments. A large range of applications based on the properties of surface plasmons has been recently developed, namely, strong light localization, elastic and inelastic scattering of light and interaction between optical and other physical properties of matter. In this topical review I address the preparation techniques of single plasmonic nanostructures, supported in homogeneous media or on substrates, and the study of their optical properties by experimental and theoretical methods. In the last decade new research directions were initiated towards metamaterials, nanoantennas, plasmonic trapping and heating. The study of quantum effects in plasmonic nanoparticles has attracted broad attention. A summary of the latest developments is presented.