Due to their low production cost, sodium-ion batteries (SIBs) are considered attractive alternatives to lithium-ion batteries (LIBs) for next generation sustainable and large-scale energy storage systems. However, during the charge/discharge cycle, a large volume strain is resulted due to the presence of a large radius of sodium ions and high molar compared to lithium ions, which further leads to poor cyclic stability and lower reversible capacity. In the past, researchers have devoted significant efforts to explore various anode materials to achieve SIBs with high energy density. Hence, as a promising anode material for SIBs, the two-dimensional (2D) materials including graphene and its derivatives and metal oxides have attracted remarkable attention due to their layered structure and superior physical and chemical properties. The inclusion of graphene and metal oxides with other nanomaterials in electrodes have led to the significant enhancements in electrical conductivity, reaction kinetics, capacity, rate performance and accommodating the large volume change respectively. Moreover, these 2D materials facilitated large surface areas and shorter paths for sodium ion adsorption and transportation respectively. In this review article, the fabrication techniques, structural configuration, sodium ion storage mechanism and its electrochemical performances will be introduced. Subsequently, an insight into the recent advancements in SIBs associated with 2D anode materials (graphene, graphene oxide (GO), transition metal oxides etc.) and other graphene-like elementary analogues (germanene, stanine etc.) as anode materials respectively will be discussed. Finally, the key challenges and future perspectives of SIBs towards enhancing the sodium storage performance of graphene-based electrode materials are discussed. In summary, we believe that this review will shed light on the path towards achieving long-cycling life, low operation cost and safe SIBs with high energy density using 2D anode materials and to be suitably commercialized for large-scale energy storage applications in the future.