A general method for calculating the (nonuniform) demagnetizing field in ferromagnetic bodies of arbitrary shape is described. The theory is based upon the assumption that the magnitude of the magnetization vector is constant throughout the sample and that its direction coincides with the direction of the local magnetic field at any point within the sample. The total magnetic field is expressed as a series of ascending powers in M/H0, where M is the saturation magnetization and H0 the applied magnetic field. The first term of this series expansion (first-order theory) gives the demagnetizing field for very large applied fields, i.e., for a uniformly magnetized sample. The higher-order corrections (we consider in detail only the first correction term; second-order theory) take account of the fact that the sample is not in general uniformly magnetized. The general theory has been applied to rectangular slabs and circular cylinders. The first-order demagnetizing field has been calculated for rectangular slabs and circular cylinders of arbitrary dimensions. Our discussion of the second-order theory is restricted to the semi-infinite slab and the semi-infinite circular cylinder. For the semi-infinite slab the variation of the second-order demagnetizing field along the central symmetry axis and across the endface have been calculated. In cases of practical interest (spin-wave propagation experiments in YIG at 3 Gc/sec and applied magnetic field of about 1400 Oe), the second-order correction to the demagnetizing factor is approximately 20% of the first-order contribution.