The gas-phase reactions of ozone with unsaturated hydrocarbons are significant sources of free radical species (including ˙OH) and particulate material in the Earth’s atmosphere. In this tutorial review, the kinetics, products and mechanisms of these reactions are examined, starting with a discussion of the original mechanism proposed by Criegee and following with a summary presentation of the complex, free radical-mediated reactions of carbonyl oxide (Criegee) intermediates. The contribution of ozone–terpene reactions to the atmospheric burden of secondary organic aerosol material is also discussed from the viewpoint of the formation of non-volatile organic acid products from the complex chemistry of ozone with α-pinene. Throughout the article, currently accepted understanding is supported through the presentation of key experimental results, and areas of persistent or new uncertainty are highlighted.