This paper gives an extensive survey on the Hamiltonians appearing in the mathematical modeling of quasicrystals. After giving an overview over results that are known to hold in arbitrary dimensions, the paper focuses mainly on discrete, one-dimensional models. Apart from surveying general spectral theory the authors discuss in detail the cases of the Fibonacci Hamiltonian and of Schrödinger operators with more general Sturmian potentials. In addition, numerical results are shown. Finally, conjectures and open problems are discussed.
For the entire collection see [Zbl 1338.37005].