In recent years, carbon-based materials, particularly carbon nanotubes (CNTs), have gained intensive research attention in the fabrication of organic solar cells (OSCs) due to their outstanding physicochemical properties, low-cost, environmental friendliness and the natural abundance of carbon. In this regard, the low sheet resistance and high optical transmittance of CNTs enables their application as alternative anodes to the widely used indium tin oxide (ITO), which is toxic, expensive and scarce. Also, the synergy between the large specific surface area and high electrical conductivity of CNTs provides both large donor-acceptor interfaces and conductive interpenetrating networks for exciton dissociation and charge carrier transport. Furthermore, the facile tunability of the energy levels of CNTs provides proper energy level alignment between the active layer and electrodes for effective extraction and transportation of charge carriers. In addition, the hydrophobic nature and high thermal conductivity of CNTs enables them to form protective layers that improve the moisture and thermal stability of OSCs, thereby prolonging the devices’ lifetime. Recently, the introduction of CNTs into OSCs produced a substantial increase in efficiency from ∼0.68 to above 14.00%. Thus, further optimization of the optoelectronic properties of CNTs can conceivably help OSCs to compete with silicon solar cells that have been commercialized. Therefore, this study presents the recent breakthroughs in efficiency and stability of OSCs, achieved mainly over 2018–2021 by incorporating CNTs into electrodes, active layers and charge transport layers. The challenges, advantages and recommendations for the fabrication of low-cost, highly efficient and sustainable next-generation OSCs are also discussed, to open up avenues for commercialization.