  We survey results that have been obtained for self-adjoint operators, and
especially Schr\"odinger operators, associated with mathematical models of
quasicrystals. After presenting general results that hold in arbitrary
dimensions, we focus our attention on the one-dimensional case, and in
particular on several key examples. The most prominent of these is the
Fibonacci Hamiltonian, for which much is known by now and to which an entire
section is devoted here. Other examples that are discussed in detail are given
by the more general class of Schr\"odinger operators with Sturmian potentials.
We put some emphasis on the methods that have been introduced quite recently in
the study of these operators, many of them coming from hyperbolic dynamics. We
conclude with a multitude of numerical calculations that illustrate the
validity of the known rigorous results and suggest conjectures for further
exploration.
