  Actions of a locally compact group G on a measure space X give rise to
unitary representations of G on Hilbert spaces. We review results on the
rigidity of these actions from the spectral point of view, that is, results
about the existence of a spectral gap for associated averaging operators and
their consequences. We will deal both with spaces X with an infinite measure as
well as with spaces with an invariant probability measure. The spectral gap
property has several striking applications to group theory, geometry, ergodic
theory, operator algebras, graph theory, theoretical computer science, etc.
