This paper presents an extensive review of the developments in the aeroelasticity of morphing aircraft that occurred in the last decade (from 2009 to 2020). It focuses mainly on fixed-wing aircraft and highlights some representative examples from rotary-wing aircraft. Morphing wings are usually associated with significant changes in the aerodynamic loads, structural/elastic properties, and inertial properties and hence the aeroelastic behavior. Changes in aeroelastic behavior can also affect the flight dynamics, stability, and control of air-vehicles. The main motivation for this paper is the fact that it is not fully possible to assess and quantify the benefits of morphing technologies without accounting for aeroelastic effects. The literature on the aeroelasticity of morphing aircraft can be split into two main themes: Aeroelastic Stability and Aeroelastic Control. The first theme (Aeroelastic Stability) includes studies conducted on morphing concepts to ensure that they satisfy certain aeroelastic requirements/constraints and that such requirements/constraints do not limit the potential benefits of morphing. The second theme (Aeroelastic Control) includes studies that utilized morphing technologies to improve aeroelastic characteristics and/or control flight loads. It is evident that in both themes, the aeroelasticity of morphing aircraft has been analyzed using analytical, numerical, and/or computational tools with a very limited number of wind-tunnel and/or flight tests. In this paper, research activities and studies are categorized according to the morphing degree of freedom they address. For each degree of freedom, research activities are arranged according to the theme they fit under. Aeroelastic frameworks developed for generic morphing applications are also reviewed. In addition, the aeroelastic models used are highlighted and discussed. Finally, trends and research gaps are identified and discussed and main conclusions are drawn.