2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139458We 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_{n-1}^{(1)}. In particular, the combinatorial approach via crystal bases and rigged configurations is discussed. Each section ends with a conglomeration of open problems.31 pages, 1 figure, axodraw and youngtab style file necessaryQuantum AlgebraCombinatorics17B37; 81R50; 82B23; 05A19text