The application of soft architectures in robotics offers the potential to reduce control complexity while increasing versatility, performance and robustness of robot operation. However, current aerial robots tend to have rigid body structures, and rely predominantly on abundant sensing and dynamic closed loop control to fly. In contrast, flying animals combine sensing and control with adaptive body designs, exploiting fluid dynamic and biomechanical effects to achieve very high levels of operational robustness and multiple functionalities. This paper provides an overview of several examples in which softness is used in aerial robotics, outlining instances where inspiration from biology can be taken to develop aerial robots which selectively use soft materials and adaptive morphologies to achieve high-performance flight behaviours. For illustration we describe three case studies where soft structures have been used in aerial robots: (1) to enable multi-modal mobility across terrain interfaces and fluid boundaries, (2) for robust perching in complex environments and (3) to repair and manufacture infrastructure components. These examples show the benefits that can be gained through the application of soft technologies and we outline how the bio-inspired approaches can be used to develop the next generation of flying robots.