Optimization of the energy levels at the donor-acceptor interface of organic solar cells has driven their efficiencies to above 10%. However, further improvements towards efficiencies comparable with inorganic solar cells remain challenging because of high recombination losses, which empirically limit the open-circuit voltage (Voc) to typically less than 1 V. Here we show that this empirical limit can be overcome using non-fullerene acceptors blended with the low band gap polymer PffBT4T-2DT leading to efficiencies approaching 10% (9.95%). We achieve Voc up to 1.12 V, which corresponds to a loss of only Eg/q - Voc = 0.5 ± 0.01 V between the optical bandgap Eg of the polymer and Voc. This high Voc is shown to be associated with the achievement of remarkably low non-geminate and non-radiative recombination losses in these devices. Suppression of non-radiative recombination implies high external electroluminescence quantum efficiencies which are orders of magnitude higher than those of equivalent devices employing fullerene acceptors. Using the balance between reduced recombination losses and good photocurrent generation efficiencies achieved experimentally as a baseline for simulations of the efficiency potential of organic solar cells, we estimate that efficiencies of up to 20% are achievable if band gaps and fill factors are further optimized. © The Royal Society of Chemistry 2016.