The wave-particle duality dates back to Einsteins explanation of the photoelectric effect through quanta of light and de Broglies 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 Wheelers 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.
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