The geometric properties of folds are considered to be of prime importance in fold classification and in the analysis of natural folds. Methods of precise description of fold geometry are necessary for natural fold analysis, and they enable a link between theoretical and experimental work and natural folds to be made. Various methods of analysing data for the spatial attitude of folds are briefly mentioned, and a detailed examination of methods of representing the geometric form of folds in profile section is made in two parts, one dealing with the forms of single layers and the other with the shapes of individual surfaces. In both cases existing methods of geometrical analysis are critically appraised and many are found to be impracticable. Folded layer geometry can most usefully be represented by two parameters that are both functions of apparent dip. These are thickness, t, and a new parameter, φ, which derives from and is used in conjunction with dip isogons. A refinement of geometrical fold classification is made in terms of these parameters, their interrelationship is examined and their relative merits are considered. In the case of single folded surfaces, a simple application of harmonic analysis provides a useful means of describing the fold shape. The most basic and suitable segment of a folded surface for analysis is a "quarter-wavelength" unit between adjacent hinge and inflexion points. Such a choice of unit leads to a harmonic series consisting only of the odd terms of a sine series. The first few harmonic coefficients are sensitive parameters of fold shape, most information being contained in the first two coefficients, b1 and b3. Plots of b3 against b1 and log bn against log n are useful means of representing and comparing coefficients for different folds. Many natural folds seem to be closely matched in shape by a member of an ideal series of mathematical forms, and this leads to a rapid visual method of harmonic analysis that involves no measurements. This has proved most useful in studies of the comparative morphology of folded rocks. Theories of fold development are discussed with particular reference to the development of folds by buckling in isolated competent layers embedded in a less competent matrix. The geometric properties of folds predicted by the theories are examined in terms of the descriptive parameters discussed in the first part of the paper. Harmonic analysis is used to describe the progressive changes in fold shape predicted by recent theory for the buckling of a layer starting as a low-amplitude sine wave. The geometric properties of passive folds are briefly considered, and the problem of "similar" folds is discussed. The geometric forms taken up by two idealised models of a buckled layer are calculated and compared. The clearest distinction between the models is brought out in the isogon patterns or by the thickness variations with dip. The effect of compression in modifying the geometry of folds is considered. Plots of thickness or φ against apparent dip for parallel folds that have been uniformly "flattened" can be so adjusted as to give straight line relationships. Most natural fold shapes are closely represented by straight lines on such adjusted graphs, and the slope or intercept of the best-fit straight line to natural fold data is an empirical parameter of folded layer shape. A simulated effect of simultaneous buckling and flattening of a layer is described that predicts relationships between thickness and dip that have been observed in natural and experimentally produced folds.