Dihydrogen complexation with retention of the H-H bond, once an exotic concept, has by now appeared in a very wide range of contexts. Three structural types are currently recognized: Kubas dihydrogen, stretched dihydrogen, and compressed dihydrides. These can be difficult to distinguish, hence the development of a number of novel spectroscopic methods for doing so, mainly based on NMR spectroscopy. Three important reactivity patterns are identified: proton loss, oxidative addition, and dissociation, each of which often contributes to larger reaction schemes, as in homogeneous hydroformylation. Main group examples are beginning to appear, although here it is mainly by computational studies that the relevant structures can be identified. Enzymes such as the hydrogenases and nitrogenases are also proposed to involve these structures.