The need for a proper selection of working materials that could withstand high-temperature environments has been the focus of recent research in the field of heat exchanger design. Heat exchangers are considered important for transferring thermal energy at different temperatures between two or more fluids to ensure efficient energy transport. Recent studies establish the current operational temperature limit to be 600oC, which highlights the need for an appropriate design and appropriate material considerations in a high-temperature condition. A typical design problem involves the large-scale deployment of renewable energy due to the scarcity of the material of choice. Despite the developments in carbon-based materials and polymers, problems arise in terms of the extraction procedure, sustainable supply, and cost-effective fabrication. In this review paper, an expansive review of the role of different materials, their fabrication compatibility for high-temperature heat exchangers (HTHXs), financial considerations, design limitations, and their general application across varied industries is presented. In addition, HTHX materials such as ceramics, carbon-based compounds, coatings, and surface treatments are presented, together with their benefits and drawbacks when selected as materials. Finally, this review paper explores current challenges, advancements, and research gaps that are available to interested researchers as a venue for their future works in the field of HTHX design and its application as a cost-effective approach.