The power conversion efficiencies (PCEs) of organic solar cells are lower than that of recently emerging perovskite solar cells. Can a PCE of >12% be achieved with single-junction organic solar cells? To achieve a high PCE, much effort has been focused on the design and synthesis of electron-donor materials, including polymers and small molecules, and on innovative solar cell device structures. In this perspective, we focus on a different approach-replacing traditional fullerene acceptors with nonfullerene organic acceptors. This method is an interesting and powerful alternative for achieving more efficient organic solar cells because the molecular structures of organic acceptors can be easily chemically modified and their optoelectronic properties and aggregation behaviors are tunable. However, the film morphology affects charge separation, transport and collection and must therefore be considered when improving the electrical performance of nonfullerene organic solar cells (NF-OSCs). Herein, we discuss molecular strategies for obtaining high-efficiency nonfullerene organic acceptors, with particular focus on small molecules. We also highlight the challenges and opportunities in this field, namely, selecting novel donor acceptor combinations, enhancing the material absorptivity to capture more solar photons, controlling the donor-acceptor interface structure to improve the charge separation, and tailoring the pi-pi-stacking structures and orientations to enhance mobile carrier transport and collection. NF-OSCs that are more efficient than traditional fullerene-based organic solar cells should be obtained by tailoring the organic acceptor structure and developing appropriate film-processing techniques, because these approaches are expected to produce NF-OSCs with a higher open-circuit voltage than their fullerene counterparts and comparable short-circuit current densities and fill factors.