Topological K-theory is one of the most important invariants for
noncommutative algebras. Bott periodicity, homotopy invariance, and
various long exact sequences distinguish it from algebraic
K-theory. This book describes a bivariant K-theory for bornological
algebras, which provides a vast generalization of topological
K-theory. In addition, it details other approaches to bivariant
K-theories for operator algebras. The book studies a number of
applications, including K-theory of crossed products, the
Baum-Connes assembly map, twisted K-theory with some of its
applications, and some variants of the Atiyah-Singer Index
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