The development of technologies to hybridise concentrating solar thermal energy (CST) and combustion technologies, is driven by the potential to provide both cost-effective CO2 mitigation and firm supply. Hybridisation, which involves combining the two energy sources within a single plant, offers these benefits over the stand-alone counterparts through the use of shared infrastructure and increased efficiency. In the near-term, hybrids between solar and fossil fuelled systems without carbon capture offer potential to lower the use of fossil fuels, while in the longer term they offer potential for low-cost carbon-neutral or carbon-negative energy. The integration of CST into CO2 capture technologies such as oxy-fuel combustion and chemical looping combustion is potentially attractive because the same components can be used for both CO2 capture and the storage of solar energy, to reduce total infrastructure and cost. The use of these hybrids with biomass and/or renewable fuels, offers the additional potential for carbon-negative energy with relatively low cost. In addition to reviewing these technologies, we propose a methodology for classifying solar-combustion hybrid technologies and assess the progress and challenges of each. Particular attention is paid to "direct hybrids", which harness the two energy sources in a common solar receiver or reactor to reduce total infrastructure and losses. (C) 2017 The Authors. Published by Elsevier Ltd.