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The requirement that the quantal wave function be single-valued is examined in the light of two recent developments: The effect of a magnetic vector potential on a particle moving in a multiply connected field-free region (Aharonov-Bohm effect), and the flux quantization inside a superconducting ring. Both of these effects are new applications of conventional elementary quantum mechanics. The single-valuedness problem is considered, with its historical background, for particles with no spin, particles with spin 12, and rigid bodies. It is shown that the single-valuedness condition, if properly adapted to the particular physical model, is deeply rooted in the foundations of quantum mechanics. However, wave functions which superficially appear to be double-valued are useful in an helicity representation of spinors and in the construction of nuclear wave functions.