The demand for larger and lighter mechanisms for next-generation space missions necessitates using deployable structures. High-strain fiber polymer composites show considerable promise for such applications due to their exceptional strength-to-weight ratio, manufacturing versatility, packaging efficiency, and capacity for self-deployment using stored strain energy. However, a significant challenge in using composite deployable structures for space applications arises from the unavoidable extended stowage periods before they are deployed into their operational configuration in orbit. During the stowage period, the polymers within the composites experience material degradation due to their inherent viscoelastic and/or plastic properties, causing stress relaxation and accumulation of plastic strains, thereby reducing the deployment capability and resulting in issues related to recovery accuracy. This paper aims to give a state-of-the-art review of recent advances in the design, modeling, and manufacturing of high-strain composites for deployable structures in space applications, emphasizing the long-term stowage effects. This review is intended to initiate discussion of future research to enable efficient, robust, and accurate design of composite deployable structures that account for the enduring challenges posed by long-term stowage effects.