Networks of organic chemical reactions are important in life and probably played a central part in its origin1,2,3. Network dynamics regulate cell division4,5,6, circadian rhythms7, nerve impulses8 and chemotaxis9, and guide the development of organisms10. Although out-of-equilibrium networks of chemical reactions have the potential to display emergent network dynamics11 such as spontaneous pattern formation, bistability and periodic oscillations12,13,14, the principles that enable networks of organic reactions to develop complex behaviours are incompletely understood. Here we describe a network of biologically relevant organic reactions (amide formation, thiolate-thioester exchange, thiolate-disulfide interchange and conjugate addition) that displays bistability and oscillations in the concentrations of organic thiols and amides. Oscillations arise from the interaction between three subcomponents of the network: an autocatalytic cycle that generates thiols and amides from thioesters and dialkyl disulfides; a trigger that controls autocatalytic growth; and inhibitory processes that remove activating thiol species that are produced during the autocatalytic cycle. In contrast to previous studies that have demonstrated oscillations and bistability using highly evolved biomolecules (enzymes15 and DNA16,17) or inorganic molecules of questionable biochemical relevance (for example, those used in Belousov-Zhabotinskii-type reactions)18,19, the organic molecules we use are relevant to metabolism and similar to those that might have existed on the early Earth. By using small organic molecules to build a network of organic reactions with autocatalytic, bistable and oscillatory behaviour, we identify principles that explain the ways in which dynamic networks relevant to life could have developed. Modifications of this network will clarify the influence of molecular structure on the dynamics of reaction networks, and may enable the design of biomimetic networks and of synthetic self-regulating and evolving chemical systems.