DNA nanotechnology makes use of the exquisite self-recognition of DNA in order to build on a molecular scale. Although static structures may find applications in structural biology and computer science, many applications in nanomedicine and nanorobotics require the additional capacity for controlled three-dimensional movement. DNA architectures can span three dimensions and DNA devices are capable of movement, but active control of well-defined three-dimensional structures has not been achieved. We demonstrate the operation of reconfigurable DNA tetrahedra whose shapes change precisely and reversibly in response to specific molecular signals. Shape changes are confirmed by gel electrophoresis and by bulk and single-molecule Förster resonance energy transfer measurements. DNA tetrahedra are natural building blocks for three-dimensional construction; they may be synthesized rapidly with high yield of a single stereoisomer, and their triangulated architecture conveys structural stability. The introduction of shape-changing structural modules opens new avenues for the manipulation of matter on the nanometre scale.