The escalating level of atmospheric carbon dioxide is one of the most pressing environmental concerns of our age. Carbon capture and storage (CCS) from large point sources such as power plants is one option for reducing anthropogenic CO2 emissions; however, currently the capture alone will increase the energy requirements of a plant by 25–40 %. This Review highlights the challenges for capture technologies which have the greatest likelihood of reducing CO2 emissions to the atmosphere, namely postcombustion (predominantly CO2/N2 separation), precombustion (CO2/H2) capture, and natural gas sweetening (CO2/CH4). The key factor which underlies significant advancements lies in improved materials that perform the separations. In this regard, the most recent developments and emerging concepts in CO2 separations by solvent absorption, chemical and physical adsorption, and membranes, amongst others, will be discussed, with particular attention on progress in the burgeoning field of metal–organic frameworks.