The demagnetizing tensor for ferromagnets is generalized to include interactions between uniformly magnetized bodies. This ''mutual'' demagnetizing tensor is symmetric, has a trace of zero, and has other simple geometric properties. The tensor is then used to develop an expression for the macroscopic magnetic field in non-uniformly magnetized bodies of arbitrary shape. Finally, the theory is applied to a block model of magnetization and explicit formulae for the tensor components are given. ¿ American Geophysical Union 1993