The current highest power-conversion efficiencies found for different types of solar cell devices range from 20% to 46%, depending on the nature of the photovoltaic materials used and device configuration. Graphene has emerged as an important organic photovoltaic material for photoenergy conversion, where graphene can be used as a transparent electrode, active interfacial layer, electron transport layer, hole transport layer, or electron/hole separation layer in fabricating solar cell devices. This review article briefly discusses some recent advances made in different types of photovoltaic materials, and then summarizes the current status of graphene-based bulk-heterojunction (BHJ) solar cells, including graphene-containing perovskite and tandem solar cell devices. Power-conversion efficiencies currently exceed 10% for heteroatom-doped multilayer graphene-based BHJ solar cells and 15.6% for graphene-containing perovskite-based solar cells. The role of graphene layer thickness, bending, thermal annealing, passivation, heteroatom doping, perovskite materials, and tandem solar cell structure on the photovoltaic performance of graphene-based solar cells is discussed. Besides aiming for high power-conversion efficiency, factors such as long-term environmental stability and degradation, and the cost-effectiveness of graphene-based solar cells for large-scale commercial production are challenging tasks.