The theory of the large elastic deformation of incompressible isotropic materials is applied to problems involving thin shells. The inflation of a circular diaphragm of such a material is studied in detail. It is found that the manner in which the extension ratios and curvatures vary in the immediate neighbourhood of the pole of the inflated diaphragm can be determined analytically. However, in order to determine their variation throughout the inflated diaphragm a method of numerical integration has to be employed. Although this is, in principle, valid for any form of the stored-energy function, the calculations are carried through only for the Mooney form. Finally, the problem of the inflation of a spherical balloon, which has already been dealt with by Green & Shield (1950), is discussed in further detail.