Birds are notable for their ability to seamlessly transition between different locomotory functions by dynamically leveraging their shape-shifting morphology. In contrast, the performance of aerial vehicles is constrained to a narrow flight envelope. To understand which functional morphological principles enable birds to successfully adapt to complex environments on the wing, engineers have started to develop biomimetic models of bird morphing flight, perching, aerial grasping and dynamic pursuit. These studies show how avian morphological capabilities are enabled by the biomaterial properties that make up their multifunctional biomechanical structures. The hierarchical structural design includes concepts like lightweight skeletons actuated by distributed muscles that shapeshift the body, informed by embedded sensing, combined with a soft streamlined external surface composed of thousands of overlapping feathers. In aerospace engineering, these functions are best replicated by smart materials, including composites, that incorporate sensing, actuation, communication, and computation. Here we provide a review of recently developed biohybrid, biomimetic, and bioinspired robot structural design principles. To inspire integrative smart material design, we first synthesize the new principles into an aerial robot concept to translate it into its aircraft equivalent. Promising aerospace applications include multifunctional morphing wing structures composed out of smart composites with embedded sensing, artificial muscles for robotic actuation, and fast actuating compliant structures with integrated sensors. The potential benefits of developing and mass-manufacturing these materials for future aerial robots and aircraft include improving flight performance, mission scope, and environmental resilience.