The analysis of shape is a key part of anatomical research and in the large majority of cases landmarks provide a standard starting point. However, while the technology of image capture has developed rapidly and in particular three‐dimensional imaging is widely available, the definitions of anatomical landmarks remain rooted in their two‐dimensional origins. In the important case of the human face, standard definitions often require careful orientation of the subject. This paper considers the definitions of facial landmarks from an interdisciplinary perspective, including biological and clinical motivations, issues associated with imaging and subsequent analysis, and the mathematical definition of surface shape using differential geometry. This last perspective provides a route to definitions of landmarks based on surface curvature, often making use of ridge and valley curves, which is genuinely three‐dimensional and is independent of orientation. Specific definitions based on curvature are proposed. These are evaluated, along with traditional definitions, in a study that uses a hierarchical (random effects) model to estimate the error variation that is present at several different levels within the image capture process. The estimates of variation at these different levels are of interest in their own right but, in addition, evidence is provided that variation is reduced at the observer level when the new landmark definitions are used.