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 features 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 how to use the bivariant Cuntz semigroup of the present work to classify unital and stably finite C⁎-algebras. © 2019 The Authors