Certain surfaces that exhibit small textured features can interact with near-wall turbulence and reduce drag, which is of great interest to industries in the aerospace, naval, transport and energy sectors. This paper reviews and discusses the dynamic mechanisms at play in that interaction. General principles of application across different technologies are discussed, and the parameters of interest and relevance are identified. It is argued that the main effect of these surfaces can be expressed as an offset between the positions of the virtual, equivalent smooth walls perceived by different parts of the flow, namely by the mean velocity profile and by the background turbulence, which remains otherwise smooth-like. Individual technologies are also reviewed, including superhydrophobic surfaces, riblets and permeable substrates, with particular emphasis on physical mechanisms that are specific to each technology. We discuss the capillary waves that form in superhydrophobic surfaces and the spanwise-elongated rollers that form over riblets and permeable surfaces.