In DNA nanotechnology, DNA molecules are designed, engineered, and assembled into arbitrary-shaped architectures with predesigned functions. Static DNA assemblies often have delicate designs with structural rigidity to overcome thermal fluctuations, whose design strategies have been studied extensively. Dynamic structures reconfigure in response to external cues. Such transformational mechanisms have been explored to create dynamic nanodevices for environmental sensing, payload delivery, and other applications. However, the precise control of reconfigurable dynamics has been a challenge due partly to flexible single-stranded DNA connections between moving parts. Deformable structures are special dynamic constructs with deformation on double-stranded parts and single-stranded hinges during reconfiguration. These structures often have better controls in programmed deformation. However, related deformability and mechanics, as well as deformation mechanisms are not well understood or documented. In this review, we summarize the development of dynamic and deformable nanostructures from the mechanics perspectives. We present deformation mechanisms such as single-stranded DNA hinges with lock-and-release pairs, jack edges, helicity modulation, and external loading. Theoretical and computational models are discussed for understanding the deformations and mechanics, including commonly used elasticity theory, finite element method, and coarse-grained molecular dynamics models. Other special models are also introduced. We elucidate the pros and cons of each model and recommend design processes based on the models. The design guidelines should be useful for those who have limited knowledge in mechanics as well as expert DNA designers. After presenting unique applications, we conclude with current challenges in dynamic and deformable structures and outlook for the development of the field.