The authors demonstrate the ability to engineer complex, twisted, and curved nanostructures from DNA through programmable self-assembly. By directing DNA strands to form tightly crosslinked double helices arrayed in parallel, they achieve controlled twist and curvature by inserting or deleting base pairs. The degree of curvature can be quantitatively controlled, with a radius of curvature as tight as 6 nanometers. They also combine multiple curved elements to create intricate nanostructures, such as wireframe beach balls and square-toothed gears. The design strategy involves stacking corrugated sheets of antiparallel helices, where each double helix has up to three nearest neighbors and is connected through anti-parallel strand crossovers. Site-directed insertions and deletions of base pairs in these arrays can control the twist and curvature of the resulting structures. The authors show that deviations from the canonical B-form DNA twist density can induce global twisting or bending, and they provide methods to quantify these effects. This work expands the design space of accessible DNA-origami shapes to include a rich diversity of nanostructures with designed twist and curvature.The authors demonstrate the ability to engineer complex, twisted, and curved nanostructures from DNA through programmable self-assembly. By directing DNA strands to form tightly crosslinked double helices arrayed in parallel, they achieve controlled twist and curvature by inserting or deleting base pairs. The degree of curvature can be quantitatively controlled, with a radius of curvature as tight as 6 nanometers. They also combine multiple curved elements to create intricate nanostructures, such as wireframe beach balls and square-toothed gears. The design strategy involves stacking corrugated sheets of antiparallel helices, where each double helix has up to three nearest neighbors and is connected through anti-parallel strand crossovers. Site-directed insertions and deletions of base pairs in these arrays can control the twist and curvature of the resulting structures. The authors show that deviations from the canonical B-form DNA twist density can induce global twisting or bending, and they provide methods to quantify these effects. This work expands the design space of accessible DNA-origami shapes to include a rich diversity of nanostructures with designed twist and curvature.