This note is devoted to the combinatorics of tableaux for the root systems $B_n$, $C_n$, $D_n$ and $G_2$ defined from Kashiwara's crystal graph theory. We review the definition of tableaux for types $B_n$, $C_n$, $D_n$ and $G_2$ and describe the corresponding bumping and sliding algorithms. We also derive in each case a Robinson-Schensted type correspondence.

We give a review of the current status of the $X=M$ conjecture. Here $X$ stands for the one-dimensional configuration sum and $M$ for the corresponding fermionic formula. There are three main versions of this conjecture: the unrestricted, the classically restricted and the level-restricted version. We discuss all three versions and illustrate the methods of proof with many examples for type $A^{(1)}_{n-1}$ . In particular, the combinatorial approach via crystal bases and rigged configurations is discussed. Each section ends with a conglomeration of open problems.

A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.