The similarities and differences between electron-nuclear tunnelling (discovered in 1966) and molecular tunnelling (discovered in 1973) are described here in terms of radiationless electron transitions. Examples of the low-temperature limit on a chemical reaction rate, observed since 1973, include: Growth of chains of polymerisation of formaldehyde, Fe-CO rebinding in haem-containing proteins, isomerisation of radical pairs, hydrobromination of ethene, and photoinduced conversion of rhodopsin into prelumirhodopsin. The general significance of molecular tunnelling for chemistry and related subjects is discussed. The possible manifestations of molecular tunnelling in the formation of complex molecules at the surfaces and in the bulk of 'dirty ice' mantles of dust grains in interstellar dense clouds are also considered. © 1979 Nature Publishing Group.