Which distributions of matter diffract? - Some answers

This review revolves around the question which general distribution of scatterers (in a Euclidean space) results in a pure point diffraction spectrum. Firstly, we treat mathematical diffration theory and state conditions under which such a distribution has pure point diffraction. We explain how a cut and project scheme naturally appears in this context and then turn our attention to the special situation of model sets and lattice substitution systems. As an example, we analyse the paperfolding sequence. In the last part, we summarize some aspects of stochastic point sets, with focus both on structure and diffraction.