Molecular self-assembly offers a `bottom-up' route to fabrication with subnanometre precision of complex structures from simple components. DNA has proved to be a versatile building block for programmable construction of such objects, including two-dimensional crystals, nanotubes, and three-dimensional wireframe nanopolyhedra. Templated self-assembly of DNA 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'. 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 100nm. 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.