The increased availability of synchrotron x-ray sources has facilitated the development of new diffraction techniques based on x-ray resonant exchange scattering (XRES). Resonant electric dipole (E1) and quadrupole (E2) scattering involving virtual transitions between occupied core and empty valence levels is sensitive to the splitting and filling of the valence shells, thus providing information about the spin and orbital distributions of the valence electrons. This sensitivity to electronic properties makes x-ray resonant exchange scattering (XRES) a useful probe of rare earth magnetism as well as effects due to crystal fields (or molecular orbitals) in transition metal compounds. In spiral antiferromagnets, such as holmium, the magnetic sensitivity results in a series of off-Bragg magnetic diffraction peaks. Resonant scattering calculations provide good predictions of the experimental observations, including diffraction conditions, intensities, lineshapes, and polarization dependence. Coefficients giving the magnitude of the scattering are computed for the rare earths. These are then related to effective scattering operators which can be expressed in terms of the angular momentum J in rare earths, making it possible to extend the results of calculations at zero temperature and zero crystal field to finite temperatures and crystal fields. Resonant scattering from transition metal ions in crystal fields includes contributions sensitive to the chemical environment of the ion. The amplitude for E1 resonances exhibits 1- and 2-fold azimuthal patterns in C\sb1 and C\sb2 symmetries respectively, but cannot distinguish C\sb3 and higher order symmetries. E2 amplitudes exhibit these patterns as well as 3- and 4-fold patterns in C\sb3 and C\sb4 symmetries. The anisotropy in the scattering from a single ion can result in diffraction at Bragg-forbidden reflections which are associated with glide plane or screw axis symmetries in the space group of the crystal. The theory provides a good description of experimental observations in hematite, including the intensity, lineshape, azimuthal pattern, Bragg-forbidden reflection, and scattered polarization.