Asymptotic Cyclic Cohomology The aim of cyclic cohomology theories is the approximation of K theory by cohomology theories defined by natural chain complexes. The basic example is the approximation of topological K theory by de Rham cohomology via the classical Chern character. A cyclic cohomology theory for operator algebras is developed in the book based on Connes work on noncommutative geometry. Asymptotic cyclic cohomology faithfully reflects the basic properties and features of operator K theory. It thus becomes a natural target for a Chern character. The central result of the book is a general Grothendieck Riemann Roch theorem in noncommutative geometry with values in asymptotic cyclic homology. Besides this the book contains numerous examples and calculations of asymptotic cyclic cohomology groups