Aircraft wings are a compromise that allows the aircraft to fly at a range of flight conditions, but the performance at each condition is sub-optimal. The ability of a wing surface to change its geometry during flight has interested researchers and designers over the years as this reduces the design compromises required. Morphing is the short form for metamorphose; however, there is neither an exact definition nor an agreement between the researchers about the type or the extent of the geometrical changes necessary to qualify an aircraft for the title ‘shape morphing.’ Geometrical parameters that can be affected by morphing solutions can be categorized into: planform alteration (span, sweep, and chord), out-of-plane transformation (twist, dihedral/gull, and span-wise bending), and airfoil adjustment (camber and thickness). Changing the wing shape or geometry is not new. Historically, morphing solutions always led to penalties in terms of cost, complexity, or weight, although in certain circumstances, these were overcome by system-level benefits. The current trend for highly efficient and ‘green’ aircraft makes such compromises less acceptable, calling for innovative morphing designs able to provide more benefits and fewer drawbacks. Recent developments in ‘smart’ materials may overcome the limitations and enhance the benefits from existing design solutions. The challenge is to design a structure that is capable of withstanding the prescribed loads, but is also able to change its shape: ideally, there should be no distinction between the structure and the actuation system. The blending of morphing and smart structures in an integrated approach requires multi-disciplinary thinking from the early development, which significantly increases the overall complexity, even at the preliminary design stage. Morphing is a promising enabling technology for the future, next-generation aircraft. However, manufacturers and end users are still too skeptical of the benefits to adopt morphing in the near future. Many developed concepts have a technology readiness level that is still very low. The recent explosive growth of satellite services means that UAVs are the technology of choice for many investigations on wing morphing. This article presents a review of the state-of-the-art on morphing aircraft and focuses on structural, shape-changing morphing concepts for both fixed and rotary wings, with particular reference to active systems. Inflatable solutions have been not considered, and skin issues and challenges are not discussed in detail. Although many interesting concepts have been synthesized, few have progressed to wing tunnel testing, and even fewer have flown. Furthermore, any successful wing morphing system must overcome the weight penalty due to the additional actuation systems.