In this chapter, we take up the study of hyperbolic n-manifolds. We begin with a geometric method for constructing spherical, Euclidean, and hyperbolic re-manifolds. In Section 11.2, we prove Poincare’s fundamental polyhedron theorem for freely acting groups. In Section 11.3, we determine the simplices of maximum volume in hyperbolic n-space. In Section 11.4, we study the Gromov invariant of a closed, orientable, hyperbolic manifold. In Section 11.5, we study the measure homology of hyperbolic space-forms. In Section 11.6, we prove Mostow’s rigidity theorem for closed, orientable, hyperbolic manifolds.