The classical examples of geometric reflection groups in Chap. 6 act on simply connected manifolds of constant curvature. This chapter concerns actions of groups generated by reflections on manifolds without geometric assumptions. In several respects such actions resemble the geometric examples. The first result, Theorem 10.1.13, is that the group W is a Coxeter group and the manifold M has the form \(M=\mathcal {U} (W,C)\) for some fundamental chamber C.