An important goal in nanotechnology is the programmable self-assembly of complex, three-dimensional nanostructures. With DNA as the building block, synthesis techniques have developed to the stage where two-dimensional designer structures and certain three-dimensional structures can be produced. Douglas et al. describe a refinement of the scaffolded DNA origami technique capable of producing three-dimensional objects of more or less any desired form, to a scale of ten to a hundred nanometres, and with an impressive degree of control over the positions of the various DNA helices. The synthesis involves DNA helices arranged on pleated strands and assembled into honeycomb-like three-dimensional structures. The various strands link together via phosphate groups. The method produces complex objects that are slow to assemble. But it also provides a route towards assembling custom devices with nanometre-scale features, as demonstrated by the construction of objects with shapes resembling a square nut, slotted cross and wire-frame icosahedron. DNA has proved to be a versatile building block in the creation of complex structures through self-assembly, exploiting the intermolecular forces between the components. Here, the arrangement of DNA helices on pleated strands which are then assembled into honeycomb-like three-dimensional structures, produces objects of unprecedented complexity. Molecular self-assembly offers a ‘bottom-up’ route to fabrication with subnanometre precision of complex structures from simple components1. DNA has proved to be a versatile building block2,3,4,5 for programmable construction of such objects, including two-dimensional crystals6, nanotubes7,8,9,10,11, and three-dimensional wireframe nanopolyhedra12,13,14,15,16,17. Templated self-assembly of DNA18 into custom two-dimensional shapes on the megadalton scale has been demonstrated previously with a multiple-kilobase ‘scaffold strand’ that is folded into a flat array of antiparallel helices by interactions with hundreds of oligonucleotide ‘staple strands’19,20. Here we extend this method to building custom three-dimensional shapes formed as pleated layers of helices constrained to a honeycomb lattice. We demonstrate the design and assembly of nanostructures approximating six shapes—monolith, square nut, railed bridge, genie bottle, stacked cross, slotted cross—with precisely controlled dimensions ranging from 10 to 100 nm. We also show hierarchical assembly of structures such as homomultimeric linear tracks and heterotrimeric wireframe icosahedra. Proper assembly requires week-long folding times and calibrated monovalent and divalent cation concentrations. We anticipate that our strategy for self-assembling custom three-dimensional shapes will provide a general route to the manufacture of sophisticated devices bearing features on the nanometre scale.