Structural DNA nanotechnology uses unusual DNA motifs to build target shapes and arrangements. These unusual motifs are generated by reciprocal exchange of DNA backbones, leading to branched systems with many strands and multiple helical domains. The motifs may be combined by sticky-ended cohesion, involving hydrogen bonding or covalent interactions. Other forms of cohesion involve edge sharing or paranemic interactions of double helices. A large number of individual species have been developed by this approach, including polyhedral catenanes, such as a cube and a truncated octahedron; a variety of single-stranded knots; and Borromean rings. In addition to these static species, DNA-based nanomechanical devices have been produced that are targeted ultimately to lead to nanorobotics. Many of the key goals of structural DNA nanotechnology entail the use of periodic arrays. A variety of two-dimensional DNA arrays have been produced with tunable features, such as patterns and cavities. DNA molecules have been used successfully in DNA-based computation as molecular representations of Wang tiles, whose self-assembly can be programmed to perform a calculation. Structural DNA nanotechnology appears to be at the cusp of a truly exciting explosion of applications, which can be expected to occur by the end of the current decade.