  We discuss discrete one-dimensional Schr\"odinger operators whose potentials
are generated by an invertible ergodic transformation of a compact metric space
and a continuous real-valued sampling function. We pay particular attention to
the case where the transformation is a minimal interval exchange
transformation. Results about the spectrum and the spectral type of these
operators are established. In particular, we provide the first examples of
transformations for which the associated Schr\"odinger operators have purely
singular spectrum for every non-constant continuous sampling function.
