Mutations of quivers were introduced by Fomin and Zelevinsky at the beginning of 2000's in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of Weyl groups. I will describe their construction and discuss its geometric interpretation: these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of small volume with proper actions of Coxeter groups. Further, the construction of presentations can be generalized to affine Weyl groups and to quotients of other Coxeter groups. This leads to a new invariant of marked surfaces. The talk is based on joint works with Anna Felikson.