This paper reviews the progress in the study of the structure of the surface of pure liquids made in the last 10-20 years. This area of research has benefited enormously in recent years from developments in experimental techniques (especially x-ray scattering methods based on third-generation synchrotron sources, and advanced optical techniques) and theory (particularly computer simulation and density functional theory). The review is predominantly experimentally based, and focuses very much on some recent exciting new results and developments. Specific systems or phenomena that are discussed are molecular ordering and orientation at the surface, atomic-scale layering at liquid metal surfaces, surface freezing in liquid alkanes and alcohols, surface melting and pre-melting at the water surface, thermal roughness (capillary waves) in simple liquids, and visco-elastic surface waves. Both the liquidvapour and liquid-solid interfaces are considered. Some details of the principal experimental techniques exploited, x-ray reflectivity, diffuse x-ray scattering, grazing incidence x-ray diffraction, ellipsometry, light scattering, will be given, The experimental developments have gone very much in conjunction with substantial theoretical advances, and the principal theoretical concepts will be briefly discussed.