Explore the rich symmetries of discrete and ultradiscrete integrable systems in this comprehensive lecture from the Workshop on "Non-commutative Geometry meets Topological Recursion" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the connections between these systems and various mathematical fields, including algebraic/tropical geometry, combinatorics, and crystal base theory. Discover how the 'tropical limit' transforms rational maps into piecewise-linear maps. Examine the symmetries of the discrete KdV equation and discrete Toda lattice, understanding their role in preserving integrability during the tropical limit. Investigate the box-ball system (BBS), an integrable cellular automata related to the aforementioned discrete systems through 'ultradiscretization.' Uncover how the BBS serves as a bridge between classical and quantum integrable systems, showcasing the intersection of tropical geometry and crystal base theory.