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 alpha-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.