Organic solar cells (OSCs) have been dominated by donor:acceptor blends based on fullerene acceptors for over two decades. This situation has changed recently, with non-fullerene (NF) OSCs developing very quickly. The power conversion efficiencies of NF OSCs have now reached a value of over 13%, which is higher than the best fullerene-based OSCs. NF acceptors show great tunability in absorption spectra and electron energy levels, providing a wide range of new opportunities. The coexistence of low voltage losses and high current generation indicates that new regimes of device physics and photophysics are reached in these systems. This Review highlights these opportunities made possible by NF acceptors, and also discuss the challenges facing the development of NF OSCs for practical applications.