  We introduce a bivariant version of the Cuntz semigroup as equivalence
classes of order zero maps generalizing the ordinary Cuntz semigroup. The
theory has many properties formally analogous to KK-theory including a
composition product. We establish basic properties, like additivity, stability
and continuity, and study categorical aspects in the setting of local
C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples
such as when the second algebra is a Kirchberg algebra, and Cuntz homology for
compact Hausdorff spaces which provides a complete invariant. Moreover, we
establish identities when tensoring with strongly self-absorbing C*-algebras.
Finally, we show that the bivariant Cuntz semigroup of the present work can be
used to classify all unital and stably finite C*-algebras.
