The authors announce proofs of certain equivariant analogues of the well- known conjectures of Novikov on higher signatures and of Borel on topological rigidity of aspherical manifolds. As a sample of their results, suppose \(f: M\to W\) is an equivariant homotopy equivalence, where G acts by isometries on a complete nonpositively curved manifold M and f is a homeomorphism outside of compact sets; then M and W are stably G-homeomorphic. They treat both smooth and Lipschitz actions of Lie groups. Their topological methods yield injectivity results for certain “assembly maps” in algebraic K-theory, L-theory, and Waldhausen’s A- theory of spaces.