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 nth 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.