Sandwich structures are a class of multifunctional high-performance structural composites that have the advantages of being lightweight, of a high strength-to-weight ratio, and of high specific energy absorption capabilities. The creative design of the core along with the apposite material selection for the fabrication of the face sheet and core are the two prerequisites with encouraging areas for further expedition towards the fabrication of advanced composite sandwich structures. The current review work focused on different types of core designs, such as truss, foam, corrugated, honeycomb, derivative, hybrid, hollow, hierarchical, gradient, folded, and smart core along with different composite materials accessible for face sheet fabrication, including fiber-reinforced composite, metal matrix composite, and polymer matrix composite are considered. The joining method plays a major role for the performance evolution of sandwich structures, which were also investigated. Further discussions are aligned to address major challenges in the fabrication of sandwich structures and further enlighten the future direction of the advanced composite sandwich structure. Finally, the work is summarized with a brief conclusion. This review article provides wider guidelines for researchers in designing and manufacturing next-generation lightweight multilayer core sandwich structures.