Abstract: Cut and project sets, which go back to Yves Meyer (1972) in mathematics and to Peter Kramer (1982) in physics, are a versatile class of structures with amazing harmonic properties. These sets are also known as mathematical quasicrystals, and include the famous Penrose tiling with fivefold symmetry as well as its various generalisations to other non-crystallographic symmetries. These constructions are widely used to model the structures discovered in 1982 by Dan Shechtman (2011 Nobel Laureate in Chemistry). More recently, also systems such as the square-free integers or the visible lattice points have been studied in this context, leading to the theory of weak model sets. This talk will review some of the developments, and introduce important concepts of the field, with focus on spectral aspects.