The mammalian cerebral cortex is a profoundly convoluted six-layered surface. The expansion of the cortex during evolution appears to be due to an increase in the number of functional units as opposed to an increase in the complexity of the units. Geometric similarity predicts that cortical area should increase in proportion to the 2/3 power of cortical volume. Allometric analysis has shown that in fact cortical area increases as a nearly linear function of cortical volume. This can be understood by appreciating that smaller brains tend to be smooth (lissencephalic) and larger brains fissured (gyrencephalic). This process of fissuration has reached its modern terrestrial limit in the human brain where the majority of the cortical surface is hidden within folds. The thickness of the cortex (2-3 mm) is small compared to its area (2000-2500 cm2) so the application of the techniques of differential geometry (the mathematics of idealized surfaces) is justified. Geometric properties of surfaces fall into two categories: intrinsic properties (which are invariant under folding of the surface, e.g. distances measured on the surface) and extrinsic properties (pure folding). The extrinsic geometry of the cortex determines the anatomical appearance of the cortex and the shape of the white matter. The intrinsic curvature of the cortex affects the relative position of functional areas and the spread of activity within the surface itself. A cortical surface has been reconstructed from cross-sections. Analysis of this surface has shown that the cortex has significant intrinsic curvature and hence it is wrong to regard it as merely a crumpled bag. The particular geometry observed is such that the surface is peculiarly "close together". Theoretical considerations and simulations suggest that the intrinsic geometry may have a significant effect on: the necessity of non-uniform growth in models of cortical development; the location of integrative areas; and the synchronization of neuronal firing. It is suggested that intrinsic descriptions of the cortex may prove more natural than extrinsic ones.