The control of structure on the nanoscale relies on intermolecular interactions whose specificity and geometry can be treated on a predictive basis. DNA fulfills this criterion, and provides an extremely convenient construction medium: The sticky-ended association of DNA molecules occurs with high specificity, and it results in the formation of double helical DNA, whose structure is well known. The use of stable branched DNA molecules permits one to make stick-figures. We have used this strategy to construct in solution a covalently closed DNA molecule whose helix axes have the connectivity of a cube: The molecule has twelve double helical edges; every edge is two helical turns in length, resulting in a hexacatenane, each of whose strands corresponds to a face of the object. We have developed a solid-support-based synthetic methodology that is more effective than solution synthesis. The key features of the technique are control over the formation of each edge of the object, and the topological closure of each intermediate. The isolation of individual objects on the surface of the support eliminates cross-reactions between growing products. The solid-support-based methodology has been used to construct a molecule whose helix axes have the connectivity of a truncated octahedron. This figure has 14 faces, of which six are square and eight are hexagonal; this Archimedean polyhedron contains 24 vertices and 36 edges, and is built from a 14-catenane of DNA. Knotted molecules appear to be the route for cloning DNA objects. It is possible to construct three knotted topologies, as well as a simple cyclic molecule from a single precursor, by control of solution conditions. Control of both branching and braiding topology is strong in this system, but control of 3-D structure remains elusive. Our key aim is the formation of prespecified 2-D and 3-D periodic structures for use in diffraction experiments. Another application envisioned is scaffolding for the assembly of molecular electronic devices.