Up until the mid 70s the kind of spectra most people had in mind in the
context of theory of Schrodinger operators were spectra occurring for
periodic potentials and for atomic and molecular Hamiltonians. Then
evidence started to build up that "exotic" spectral phenomena such as
singular continuous, Cantor, and dense point spectrum do occur in
mathematical models that are of substantial interest to theoretical
physics. One area where such exotic phenomena are particularly abundant is
quasiperiodic operators. They feature a competition between
randomness (ergodicity) and order (periodicity), which is often resolved
at a deep arithmetic level. Mathematically, the methods involved include a
mixture of ergodic theory, dynamical systems, probability, functional and
harmonic analysis and analytic number theory. The interest in those models was enhanced by strong
connections with some major discoveries in physics, such as integer
quantum Hall effect, experimental quasicrystals, and quantum chaos theory,
in all of which quasiperiodic operators provide central or important
models.

We will give a general overview concentrating on aspects where the
competition and/or collaboration between order and chaos plays an
important role