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The box-ball system is an integrable cellular automaton on one dimensionallattice. It arises from either quantum or classical integrable systems by theprocedures called crystallization and ultradiscretization, respectively. Thedouble origin of the integrability has endowed the box-ball system with avariety of aspects related to Yang-Baxter integrable models in statisticalmechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz,geometric crystals, classical theory of solitons, tau functions, inversescattering method, action-angle variables and invariant tori in completelyintegrable systems, spectral curves, tropical geometry and so forth. In thisreview article, we demonstrate these integrable structures of the box-ballsystem and its generalizations based on the developments in the last twodecades.

Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry