DNA nanotechnology provides an excellent foundation for diverse nanoscale structures that can be used in various bioapplications and materials research. Among all existing DNA assembly techniques, DNA origami proves to be the most robust one for creating custom nanoshapes. Since its invention in 2006, building from the bottom up using DNA advanced drastically, and therefore, more and more complex DNA-based systems became accessible. So far, the vast majority of the demonstrated DNA origami frameworks are static by nature; however, there also exist dynamic DNA origami devices that are increasingly coming into view. In this review, we discuss DNA origami nanostructures that exhibit controlled translational or rotational movement when triggered by predefined DNA sequences, various molecular interactions, and/or external stimuli such as light, pH, temperature, and electromagnetic fields. The rapid evolution of such dynamic DNA origami tools will undoubtedly have a significant impact on molecular-scale precision measurements, targeted drug delivery and diagnostics; however, they can also play a role in the development of optical/plasmonic sensors, nanophotonic devices, and nanorobotics for numerous different tasks.