Shape memory polymers and their composites are kind of smart materials, which can transfer from the temporarily fixed configuration to the original configuration under external stimuli. Their inherent advantages are low density, low cost, large recoverable deformation ability and controllable stimulus method, which make them an alternative for aerospace applications (deployable structures, release devices, wrinkled/slack control component, etc). Most of these applications are in development stage, some have completed the ground functional verification experiments, a rare part has been carried out the spaceflight experiments. This review focuses on their materials and structures in aerospace field, briefly introduces the development history and general mechanism of shape memory polymers and their composites, replenishes the space radiation resistance abilities of these materials under the ground-simulated space radiation experiments, tracks those applications who have already completed the spaceflight experiments, then exhibits some novel applications which have great potential and could give us inspirations for new aerospace application development, finally the prospects for materials and structures are discussed in the future outlook.