The energy transfer between adjacent resonances in nuclear and electron spin systems is analyzed in terms of the overlap of line-shape functions. The procedure is an enlargement on the original proposal of Kronig and Bouwkamp, and consists of taking partial account of off-diagonal elements in the spin-spin interaction, which are omitted in Van Vleck's truncated Hamiltonian. If the frequency of these off-diagonal elements is sufficiently small, they give rise to an additional kind of spin-spin relaxation, observed by Gorter and co-workers. They are also responsible for cross-saturation effects in paramagnetic salts of the type observed by Townes and co-workers. A crucial experiment is described which can be explained by spin-spin interactions, but not by the assumption of a hot-phonon region. Implications of the cross-relaxation for the operation of solid state masers are discussed. Special consideration is given to magnetically dilute substances and inhomogeneously broadened lines. Paradoxically, the latter will usually still undergo a homogeneous steady-state saturation.