In 1984 Shechtman, Blech, Gratias and Cahn discovered solids with a very distinctive form of disorder, later called "aperiodic order": The diffraction pattern of such a solid is pure point indicating a high degree of order and at the same time exhibits symmetries which exclude periodicity. In this context, the study of pure point diffraction has attracted much attention in recent years. It turns out that pure point diffraction is closely related to pure point spectrum of certain naturally associated dynamical systems. We will report on recent results on this relationship. This is joint work with Michael Baake.