Abstract Morphing, understood as the ability to undergo pronounced shape adaptations to optimally respond to a diversity of operational conditions, has been singled out as a future direction in the pursuit of maximised efficiency of lightweight structures. Whereas a certain degree of adaptivity can be accomplished conventionally by means of mechanical systems, compliance allowing for substantial reversible deformability exhibits far more potential as a morphing strategy. A promising solution to the inherent contradiction between high stiffness and reversible deformation capacity posed by morphing is offered by introducing variable stiffness components. This notion indicates the provision of a controllable range of deformation resistance levels in place of fixed properties, as required by real-time shape adaptation dictated by maximum efficiency under changing external conditions. With special emphasis on the morphing context, the current review aims to identify the main tendencies, undertaking a systematic classification of existing approaches involving stiffness variability. Four broad categories in which variable stiffness has been applied to morphing are therefore distinguished and detailed: material engineering, active mechanical design, semi-active techniques and elastic structural behaviour. Adopting a wide perspective, the study highlights key capabilities, limitations and challenges. The need for attention directed to the variable stiffness strategy is recognised and the significance of intensive research activities in a highly integrated and multidisciplinary environment emphasised if higher maturity stages of the concepts are to be reached. Finally, the potential of emerging directions of semi-active design involving electro-bonded laminates and multi-stable structures is brought into focus.