Higson-Roe compactifications first arose in connection with C^*-algebra approaches to index theory on noncompact manifolds. Vanishing and/or equivariant splitting results for the cohomology of these compactifications imply the integral Novikov Conjecture for fundamental groups of finite aspherical CW complexes. We survey known results on these compactifications and prove some new results -- most notably that the n^th cohomology of the Higson-Roe compactification of hyperbolic space Hⁿ consists entirely of 2-torsion for n even and that the rational cohomology of the Higson-Roe compactification of Rⁿ is nontrivial in all dimensions les than or equal to n.