Nowadays, rechargeable batteries are widely used for many different purposes and they are among the main clean energy storage systems that are readily available. Electrical energy is stored in the battery through oxidation and reduction and via a charge transfer agent that moves between the cathode and anode. Rechargeable batteries commonly use the alkali metal, lithium (Li) as the charge transfer agent. The capacity of the anode material and selection of a charge transfer agent that is easily adsorbed by the anode material are two important factors when designing rechargeable batteries. In this PhD thesis, we propose several types of carbonaceous materials that could be potentially used as an anode in rechargeable batteries. These materials are derivatives of graphene: graphdiyne (GDY), hydrogenated defective graphene (H-DG), edge-functionalised graphene nanoribbons (F-GNRs), and doped graphene. In addition, we select the charge transfer agents, sodium (Na) and calcium (Ca) on several materials, and we also consider the use of potassium (K). The first material, GDY, adsorbed Na atoms with a binding energy appropriate for battery applications. Adsorption occurs on the 6-membered ring and the pores, with an empirical formula of NaC2.57, which is equivalent to an electrical capacity of 497 mA h g-1. If expansion is allowed for the bulk layered GDY, this loading will occur on each layer of bulk GDY, with an expansion of 28% of the interlayer spacing compared with the unloaded material. Furthermore, the bulk layered GDY in an AB-2 stacking has a barrier of energy against Na movement, parallel to the layers of 0.84 eV and as low as 0.12 eV for vertical movement within a slit pore. These values were even lower for the bulk material with AB-3 stacking. A hydrogenated mono-vacancy repeated four times within a graphene sheet supercell with 68 carbon atoms, 4(H1-MVG), has been considered as another suitable material for the NIB and CIBs. It also provided a model hydrogenated graphene that could be compared with graphene. Compared with graphene, this material strengthened the binding of Na by an energy of 0.72 eV. A similar strengthening occurred for Ca binding. This material can bind up to 16 Na and 14 Ca atoms per 4(H1-MVG) supercell on a single layer. Although the bulk layered 4(H1-MVG) did not strongly bind Na within the layers, Ca effectively intercalated within the layers, with binding energies varying between-2.05 to-2.79 eV, which is strong in comparison with Ca binding to graphene (-0.82 eV). Finally, we found that up to 16 Ca atoms could intercalate ~ ii ~ within the bulk layered material, which is equivalent to an electrical capacity of 591.2 m A h g-1 and results in 29.3% for the interlayer expansion. Edge-functionalised graphene nanoribbons have been proposed as the third material for NIBs and CIBs. Nanoribbons with zigzag and armchair edges are functionalised with the selected oxygen-containing functional groups hydroxyl (HO-), carbonyl (O=), and carboxyl (HOOC-). These groups are found abundantly in reduced g...