An assortment of carbon-based materials, such as nanotubes, nanorods, nanoribbons, nanofibers and graphene, is fast gaining significant research interest in developing various components of organic solar cells (OSCs) due to their unique optoelectronic properties. Among these, graphene-based materials are more appealing owing to their remarkable optical, electrical, chemical, mechanical and thermal properties, coupled with their specific large surface area and flexibility, which are compatible with large-scale roll-to-roll synthesis. Their low-cost, abundance, non-toxicity, high optical transparency and competitive electrical conductivity makes them potential replacement materials for the commonly used indium tin oxide (ITO) anodes in bulk heterojunction (BHJ)-OSCs owing to the scarcity, high cost and toxicity of indium, which is the principal constituent element of ITO. Furthermore, the synergy between graphene-based electron-acceptor materials and donor polymers in the photoactive layer of BHJ-OSCs results in enhanced photon harvesting, improved exciton generation, effective exciton dissociation and efficient charge transport. However, graphene-based materials have been applied not only as anodes and electron acceptors but also as cathodes, electron-transport layers and hole transport layers. Recently, the incorporation of graphene-based materials into OSCs has led to a significant increase in power conversion efficiency (PCE) from similar to 0.63% to above 16.00%. The PCE can be further enhanced by employing recent breakthroughs to optimize the optoelectronic properties of the various OSC components; hence making graphene-based state-of-the-art OSCs approach PCEs of similar to 25%, which approach the favourable PCE values of above 26% exhibited by silicon-based solar cells. Thus, graphene-based OSCs can conceivably close the gap between OSCs and silicon-based devices. Herein, we present an in-depth review of the recent progress on applying graphene-based materials in BHJ-OSCs as electrodes, electron acceptors and interfacial layers, for the advancement and realization of high-efficiency and sustainable devices, as a link-bridge towards commercialization. The crucial photovoltaic parameters, such as short-circuit current density, open-circuit voltage, fill factor and PCE, are discussed to reveal the merits, drawbacks and future prospects of graphene-based BHJ-OSCs.