  Kinematic diffraction is well suited for a mathematical approach via
measures, which has substantially been developed since the discovery of
quasicrystals. The need for further insight emerged from the question of which
distributions of matter, beyond perfect crystals, lead to pure point
diffraction, hence to sharp Bragg peaks only. More recently, it has become
apparent that one also has to study continuous diffraction in more detail, with
a careful analysis of the different types of diffuse scattering involved. In
this review, we summarise some key results, with particular emphasis on
non-periodic structures. We choose an exposition on the basis of characteristic
examples, while we refer to the existing literature for proofs and further
details.
