Typically, polymeric materials experience material degradation and damage over time in harsh environments. Improved understanding of the physical and chemical processes associated with possible damage modes intended in high-pressure hydrogen gas exposed atmospheres will help to select and develop materials well suited for applications fulfilling future energy demands in hydrogen as an energy carrier. In high-pressure hydrogen gas exposure conditions, damage from rapid gas decompression (RGD) and from aging in elastomeric as well as thermoplastic material components is unavoidable. This review discusses the applications of polymeric materials in a multi-material approach in the realization of the “Hydrogen economy”. It covers the limitations of existing polymeric components, the current knowledge on polymeric material testing and characterization, and the latest developments. Some improvements are suggested in terms of material development and testing procedures to fill in the gaps in existing knowledge in the literature.