2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75603By a recent result obtained by R. Howlett and the author considerable progress has been made towards a complete solution of the isomorphism problem for Coxeter groups. In this paper we give a survey on the isomorphism problem and explain in particular how the result mentioned above reduces it to its `reflection preserving' version. Furthermore we desrcibe recent developments concerning the solution of the latter.15 pages, 0 figures, to appear in 'The Coxeter Legacy: Reflections and Projections', Fields Institute CommunicationsGroup Theory20F55 (primary), 51F55 (secondary)text