Discrete and ultradiscrete integrable systems have rich symmetries related to various mathematics: algebraic/tropical geometry, combinatorics, crystal base theory and so on. These systems are related by a `tropical limit' which sends rational maps to piecewise-linear maps.
In this lecture, first we introduce the symmetries of discrete KdV equation and discrete Toda lattice. They have nice structures which make it possible to preserve the integrability in the tropical limit. Next we discuss an integrable cellular automata called the box-ball system (BBS) related to the two discrete systems via `ultradiscretization (a tropical limit restricted on integers)'. The BBS is on a cross-point of classical and quantum integrable systems, where tropical geometry (a tropical limit of algebraic geometry) and crystal base theory (a limit of the quantum group) meet.