This review covers the structure of, and chemical reactions on, well-defined cerium oxide surfaces. Ceria, or mixed oxides containing ceria, are critical components in automotive threeway catalysts due to their well-known oxygen storage capacity. Ceria is also emerging as an important material in a number of other catalytic processes, particularly those involving organic oxygenates and the water-gas shift reaction. Ceria's acid-base properties, and thus its catalytic behavior, are closely related to its surface structure where different oxygen anion and cerium cation environments are present on the low-index structural faces. The actual structure of these various faces has been the focus of a number of theoretical and experimental investigations. Ceria is also easily reducible from CeO 2 to CeO 2-X. The presence of oxygen vacancies on the surface often dramatically alters the adsorption and subsequent reactions of various adsorbates, either on a clean surface or on metal particles supported on the surface. Most surface science studies have been conducted on the surfaces of thin-films rather than on the surfaces of bulk single crystal oxides. The growth, characterization and properties of these thinfilms are also examined.