The wave-particle duality dates back to Einstein's explanation of the
photoelectric effect through quanta of light and de Broglie's hypothesis of
matter waves. Quantum mechanics uses an abstract description for the behavior
of physical systems such as photons, electrons, or atoms. Whether quantum
predictions for single systems in an interferometric experiment allow an
intuitive understanding in terms of the particle or wave picture, depends on
the specific configuration which is being used. In principle, this leaves open
the possibility that quantum systems always either behave definitely as a
particle or definitely as a wave in every experimental run by a priori adapting
to the specific experimental situation. This is precisely what is tried to be
excluded by delayed-choice experiments, in which the observer chooses to reveal
the particle or wave character -- or even a continuous transformation between
the two -- of a quantum system at a late stage of the experiment. We review the
history of delayed-choice gedanken experiments, which can be traced back to the
early days of quantum mechanics. Then we discuss their experimental
realizations, in particular Wheeler's delayed choice in interferometric setups
as well as delayed-choice quantum erasure and entanglement swapping. The latter
is particularly interesting, because it elevates the wave-particle duality of a
single quantum system to an entanglement-separability duality of multiple
systems
