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DNA nanotechnology has been developed in order to construct nanostructures and nanomachines by virtue of the programmable self-assembly properties of DNA molecules. Although DNA nanotechnology initially focused on spatial arrangement of DNA strands, new horizons have been explored owing to the development of the toehold-mediated strand-displacement reaction, conferring new dynamic properties to previously static and rigid structures. A large variety of DNA reconfigurable nanostructures, stepped and autonomous nanomachines and circuits have been operated using the strand-displacement reaction. Biological systems rely on information processing to guide their behaviour and functions. Molecular computation is a branch of DNA nanotechnology that aims to construct and operate programmable computing devices made out of DNA that could interact in a biological context. Similar to conventional computers, the computational processes involved are based on Boolean logic, a propositional language that describes statements as being true or false while connecting them with logic operators. Numerous logic gates and circuits have been built with DNA that demonstrate information processing at the molecular level. However, development of new systems is called for in order to perform new tasks of higher computational complexity and enhanced reliability. The contribution of secondary structure to the vulnerability of a toehold-sequestered device to undesired triggering of inputs was examined, giving new approaches for minimizing leakage of DNA devices. This device was then integrated as a logic component in a DNA-based computer with a retrievable memory, thus implementing two essential biological functions in one synthetic device. Additionally, G-quadruplex logic gates were developed that can be switched between two topological states in a logic fashion. Their individual responses were detected simultaneously, establishing a new approach for parallel biological computing. A new AND-NOT logic circuit based on the seesaw mechanism was constructed that, in combination with the already existing AND and OR gates, form a now complete basis set that could perform any Boolean computation. This work introduces a new mode of kinetic control over the operation of such DNA circuits. Finally, the first example of a transmembrane logic gate being operated at the single-molecule level is described. This could be used as a potential platform for biosensing.