Exploiting fluid-structure interactions using flexible structures and actuation can improve performance in biologically inspired swimming and morphing airfoils. Open challenges in this realm are the efficient high-fidelity simulation of such fluid-structure interaction phenomena, especially in three-dimensions, as well as solving inverse problems for optimal design and control. Compounding the latter challenge in robotic design is the high-dimensional structure-actuation parameter space facilitated by modern developments in smart materials and structures, which prohibits brute-force optimization or control approaches. In this talk I will discuss our recent work on both forward and inverse problems in morphing structures and fluid-structure interaction, combining theoretical, numerical, and data-driven machine learning approaches. Specifically, I will first highlight our progress on 3D simulations of fluid flows with moving/deforming boundaries, using high-order immersed methods, high-order grid adaptation techniques, and large-scale parallel computing. Second, I will talk about how we use numerical simulations to train data-driven surrogate problems for solving inverse problems in flexible flapping foil propulsion.