Summary: We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of M.-P. Schützenberger [Lect. Notes Math. 579, 59–113 (1977; Zbl 0398.05011)] for standard Young tableaux. We apply this to give a new combinatorial rule for the \(K\)-theory Schubert calculus of Grassmannians via \(K\)-theoretic jeu de taquin, providing an alternative to the rules of Buch and others. This rule naturally generalizes to give a conjectural root-system uniform rule for any minuscule flag variety \(G/P\), extending recent work of Thomas and Yong. We also present analogues of results of Fomin, Haiman, Schensted and Schützenberger.