We define and characterize switching, an operation that takes
two Young tableaux sharing a common border, ``moves them through each
other'', and produces another such pair. This operation clarifies a
connection between algorithms described by Haiman and by James and Kerber.
Haiman's algorithm is a generalization of Schützenberger's jeu de
taquin, while the algorithm of James and Kerber has been used by White to
prove a generalization of the Littlewood-Richardson rule. We establish new
results and provide new proofs of results concerning the jeu de
taquin, evacuation, Schur functions, Young tableaux, characters of
representations, branching rules, the Littlewood-Richardson rule, and
symmetries of Littlewood-Richardson coefficients.