Morphing structures change their geometric configuration to achieve a wide range of performance goals. For morphing aircraft these include alleviating drag, or altering aerofoil lift. The design of structures capable of realising these goals is a highly multidisciplinary problem. Optimally morphing a compliant structure involves finding the distribution of actuation which best achieves a desired configuration change. In this work, the location and magnitude of discrete actuators are optimised, to minimise both aerodynamic and geometric objective functions. A range of optimisation methods, including differential and stochastic techniques, has been implemented to search optimally the large, nonlinear, and often discontinuous design spaces associated with such problems. The optimal design of morphing systems is investigated through consideration of a morphing shock control bump and an adaptive leading edge. CFD is implemented to evaluate the aerodynamic performance of optimiser-controlled morphing structures. A bespoke grid-generation algorithm is developed, capable of producing a mesh for all possible geometries, with low levels of cell skewness and orthogonality at the fluid-structure boundaries. Structural compliance – a prerequisite for morphing – allows significant displacement of the structure to occur, but simultaneously enables the possibility of detrimental aeroelastic effects. Static aeroelasticity is catered for, at significant computational expense, via coupling of the structural and aerodynamic models within individual optimisation function evaluations. Morphing geometry is investigated to reduce computational design requirements, and provide an objective starting point for an aeroelastic optimisation. The requirements of morphing between aerodynamic shapes are evaluated using geometry-based objective functions. Displacements and curvatures are compared between an optimiser-controlled structure and the target morph, and the differences minimised to effect the required shape change. In addition to enabling optimal problem definition, these geometric objective functions allow conclusions on the feasibility of a morph to be drawn a priori.