  We discuss plane wave backgrounds of string theory and their relation to
Goedel-like universes. This involves a twisted compactification along the
direction of propagation of the wave, which induces closed timelike curves. We
show, however, that no such curves are geodesic. The particle geodesics and the
preferred holographic screens we find are qualitatively different from those in
the Goedel-like universes. Of the two types of preferred screen, only one is
suited to dimensional reduction and/or T-duality, and this provides a
``holographic protection'' of chronology. The other type of screen, relevant to
an observer localized in all directions, is constructed both for the compact
and non-compact plane waves, a result of possible independent interest. We
comment on the consistency of field theory in such spaces, in which there are
closed timelike (and null) curves but no closed timelike (or null) geodesics.
