The tensorial interactions of X-rays in crystals are reviewed here, including magnetic effects, both in X-ray absorption and X-ray scattering. After a brief historical description of the development of theoretical concepts and experimental work in the last two decades of the twentieth century and the first decade of the twenty-first century, their analysis is detailed. The basic theoretical formulae are provided with the main information about their origin and a necessary link to the original literature in order to analyze them in more depth. Two examples of the experimental detection of higher-order magnetoelectric multipoles are provided: in V2O3, using resonant elastic scattering techniques, and in GaFeO3, using X-ray absorption-based techniques.