Morphing aircraft can adaptively regulate their aerodynamic layout to meet the demands of varying flight conditions, improve their aerodynamic efficiency, and reduce their energy consumption. The design and fabrication of high-performance, lightweight, and intelligent morphing structures have become a hot topic in advanced aircraft design. This paper discusses morphing aircraft development history, structural characteristics, existing applications, and future prospects. First, some conventional mechanical morphing aircraft are examined with focus on their morphing modes, mechanisms, advantages, and disadvantages. Second, the novel applications of several technologies for morphing unmanned aerial vehicles, including additive manufacturing for fabricating complex morphing structures, lattice technology for reducing structural weight, and multi-mode morphing combined with flexible skins and foldable structures, are summarized and categorized. Moreover, in consideration of the further development of active morphing aircraft, the paper reviews morphing structures driven by smart material actuators, such as shape memory alloy and macro-fiber composites, and analyzes their advantages and limitations. Third, the paper discusses multiple challenges, including flexible structures, flexible skins, and control systems, in the design of future morphing aircraft. Lastly, the development and application of morphing structures in the aerospace field are discussed to provide a reference for future research and engineering applications.