Plasmonic structures can be constructed from precise numbers of well-defined metal nanoparticles that are held together with molecular linkers, templates or spacers. Such structures could be used to concentrate, guide and switch light on the nanoscale in sensors and various other devices. DNA was first used to rationally design plasmonic structures in 1996, and more sophisticated motifs have since emerged as effective and versatile species for guiding the assembly of plasmonic nanoparticles into structures with useful properties. Here we review the design principles for plasmonic nanostructures, and discuss how DNA has been applied to build finite-number assemblies (plasmonic molecules), regularly spaced nanoparticle chains (plasmonic polymers) and extended two- and three-dimensional ordered arrays (plasmonic crystals).