This book presents a systematic study of a new equivariant cohomology theory \(t(k_G)^*\) constructed from any given equivariant cohomology theory \(k^*_G\), where \(G\) is a compact Lie group. Special cases include Tate-Swan cohomology when \(G\) is finite and a version of cyclic cohomology when \(G = S^1\). The groups \(t(k_G)^*(X)\) are obtained by suitably splicing the \(k\)-homology with the \(k\)-cohomology of the Borel construction \(EG\times _G X\), where \(k^*\) is the nonequivariant cohomology theory that underlies \(k^*_G\). The new theories play a central role in relating equivariant algebraic topology with current areas of interest in nonequivariant algebraic topology. Their study is essential to a full understanding of such “completion theorems” as the Atiyah-Segal completion theorem in \(K\)-theory and the Segal conjecture in cohomotopy. When \(G\) is finite, the Tate theory associated to equivariant \(K\)-theory is calculated completely, and the Tate theory associated to equivariant cohomotopy is shown to encode a mysterious web of connections between the Tate cohomology of finite groups and the stable homotopy groups of spheres.