Given a Coxeter system \((W,S)\) , a space X and a family of subspaces \((X_s)_{s\in S}\) , there is a classical construction of a space \(\mathcal {U} (W,X)\) with W-action. \(\mathcal {U}(W,X)\) is constructed by pasting together copies of X, one for each element of W. The purpose of this chapter is to give the details of this construction. More generally, in Sect. 5.1 we describe the construction for an arbitrary group G together with a “family of subgroups.” This greater generality will not be needed until Chap. 20 when it will be used in the discussion of geometric realizations of buildings.