Space exploration requires the use of suitable materials to protect astronauts and structures from the hazardous effects of radiation, in particular, ionizing radiation, which is ubiquitous in the hostile space environment. In this scenario, polymer-based materials and composites play a crucial role in achieving effective radiation shielding while providing low-weight and tailored mechanical properties to spacecraft components. This work provides an overview of the latest developments and challenges in polymer-based materials designed for radiation-shielding applications in space. Recent advances in terms of both experimental and numerical studies are discussed. Different approaches to enhancing the radiation-shielding performance are reported, such as integrating various types of nanofillers within polymer matrices and optimizing the materials design. Furthermore, this review explores the challenges in developing multifunctional materials that are able to provide radiation protection. By summarizing the state-of-the-art research and identifying emerging trends, this review aims to contribute to the ongoing efforts to identify polymer materials and composites that are most useful to protect human health and spacecraft performance in the harsh radiation conditions that are typically found during missions in space.