Nuclear energy offers a wide range of applications, which include power generation, X-ray imaging, and non-destructive tests, in many economic sectors. However, such applications come with the risk of harmful radiation, thereby requiring shielding to prevent harmful effects on the surrounding environment and users. Concrete has long been used as part of structures in nuclear power plants, X-ray imaging rooms, and radioactive storage. The direction of recent research is headed toward concrete’s ability in attenuating harmful energy radiated from nuclear sources through various alterations to its composition. Radiation shielding concrete (RSC) is a composite-based concrete that was developed in the last few years with heavy natural aggregates such as magnetite or barites. RSC is deemed a superior alternative to many types of traditional normal concrete in terms of shielding against the harmful radiation, and being economical and moldable. Given the merits of RSCs, this article presents a comprehensive review on the subject, considering the classifications, alternative materials, design additives, and type of heavy aggregates used. This literature review also provides critical reviews on RSC performance in terms of radiation shielding characteristics, mechanical strength, and durability. In addition, this work extensively reviews the trends of development research toward a broad understanding of the application possibilities of RSC as an advanced concrete product for producing a robust and green concrete composite for the construction of radiation shielding facilities as a better solution for protection from sources of radiation. Furthermore, this critical review provides a view of the progress made on RSCs and proposes avenues for future research on this hotspot research topic.