Recent advances in biostereometric techniques have led to the quick and easy acquisition of 3D data for facial and other biological surfaces. This has led facial surgeons to express dissatisfaction with landmark-based methods for analysing the shape of the face which use only a small part of the data available, and to seek a method for analysing the face which maximizes the use of this extensive data set. Scientists working in the field of computer vision have developed a variety of methods for the analysis and description of 2D and 3D shape. These methods are reviewed and an approach, based on differential geometry, is selected for the description of facial shape. For each data point, the Gaussian and mean curvatures of the surface are calculated. The performance of three algorithms for computing these curvatures are evaluated for mathematically generated standard 3D objects and for 3D data obtained from an optical surface scanner. Using the signs of these curvatures, the face is classified into eight 'fundamental surface types' - each of which has an intuitive perceptual meaning. The robustness of the resulting surface type description to errors in the data is determined together with its repeatability. Three methods for comparing two surface type descriptions are presented and illustrated for average male and average female faces. Thus a quantitative description of facial change, or differences between individual's faces, is achieved. The possible application of artificial intelligence techniques to automate this comparison is discussed. The sensitivity of the description to global and local changes to the data, made by mathematical functions, is investigated. Examples are given of the application of this method for describing facial changes made by facial reconstructive surgery and implications for defining a basis for facial aesthetics using shape are discussed. It is also applied to investigate the role played by the shape of the surface in facial recognition.