Structural DNA nanotechnology, in which Watson-Crick base pairing drives the formation of self-assembling nanostructures, has rapidly expanded in complexity and functionality since its inception in 1981. DNA nanostructures can now be made in arbitrary three-dimensional shapes and used to scaffold many other functional molecules such as proteins, metallic nanoparticles, polymers, fluorescent dyes and small molecules. In parallel, the field of dynamic DNA nanotechnology has built DNA circuits, motors and switches. More recently, these two areas have begun to merge—to produce switchable DNA nanostructures, which change state in response to their environment. In this review, we summarise switchable DNA nanostructures into two major classes based on response type: molecular actuation triggered by local chemical changes such as pH or concentration and external actuation driven by light, electric or magnetic fields. While molecular actuation has been well explored, external actuation of DNA nanostructures is a relatively new area that allows for the remote control of nanoscale devices. We discuss recent applications for DNA nanostructures where switching is used to perform specific functions—such as opening a capsule to deliver a molecular payload to a target cell. We then discuss challenges and future directions towards achieving synthetic nanomachines with complexity on the level of the protein machinery in living cells.