  The box-ball system is an integrable cellular automaton on one dimensional
lattice. It arises from either quantum or classical integrable systems by the
procedures called crystallization and ultradiscretization, respectively. The
double origin of the integrability has endowed the box-ball system with a
variety of aspects related to Yang-Baxter integrable models in statistical
mechanics, crystal base theory in quantum groups, combinatorial Bethe ansatz,
geometric crystals, classical theory of solitons, tau functions, inverse
scattering method, action-angle variables and invariant tori in completely
integrable systems, spectral curves, tropical geometry and so forth. In this
review article, we demonstrate these integrable structures of the box-ball
system and its generalizations based on the developments in the last two
decades.
