While the isomorphism problem is undecidable in the full universe of groups, the region containing Coxeter groups is fairly well understood and it is expected that the problem is solvable within this class of groups. The Coxeter galaxy provides a framework to study this and related, more refined questions. In this talk we will provide an overview on the current state of the art concerning this problem and present some new results obtained in joint work with Yuri Santos Rego.