The author develops theoretical methods applicable to light scattering in dispersive absorbing crystals. Special attention is paid to scattering with widely different absorption at incident and scattered frequencies that generates frequency shifted radiation (hyperresonance). Different types of scattering are considered: Raman scattering, hyper-Raman and hyper-Rayleigh scattering, and stimulated Raman scattering. Experimental data on scattering in crystalline powders, films, single crystals, and near-surface crystal layers are analyzed.