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Organic compounds are ubiquitous in the Earth's surface, sediments and many rocks, and preserve records of geological, geochemical and biological history; they are also critical natural resources and major environmental pollutants. The naturally occurring stable isotopes of volatile elements (D, ^(13)C, ^(15)N, ^(17,18)O, ^(33,34,36)S) provide one way of studying the origin, evolution and migration of geological organic compounds. The study of bulk stable isotope compositions (i.e. averaged across all possible molecular isotopic forms) is well established and widely practised, but frequently results in non-unique interpretations. Increasingly, researchers are reading the organic isotopic record with greater depth and specificity by characterizing stable isotope ‘structures’ – the proportions of site-specific and multiply substituted isotopologues that contribute to the total rare-isotope inventory of each compound. Most of the technologies for measuring stable isotope structures of organic molecules have been only recently developed and to date have been applied only in an exploratory way. Nevertheless, recent advances have demonstrated that molecular isotopic structures provide distinctive records of biosynthetic origins, conditions and mechanisms of chemical transformation during burial, and forensic fingerprints of exceptional specificity. This paper provides a review of this young field, which is organized to follow the evolution of molecular isotopic structure from biosynthesis, through diagenesis, catagenesis and metamorphism.