Structural DNA nanotechnology provides an excellent foundation for diverse nanoscale shapes that can be used in various bioapplications and materials research. From all existing DNA assembly techniques, DNA origami has proven to be the most robust one for creating custom nanoshapes. Since its invention in 2006, building from the bottom up using DNA has drastically advanced, and therefore, more and more complex DNA-based systems have become accessible. So far, vast majority of the demonstrated DNA origami frameworks are static by nature, but interestingly, there also exist dynamic DNA origami devices that are increasingly coming into view. In this review, we discuss DNA origami nanostructures that perform controlled translational or rotational movement triggered by predefined DNA strands, various molecular interactions and/or other 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, but they can also play a role in development of optical/plasmonic sensors, nanophotonic devices and nanorobotics for numerous different tasks.