In 1990, Connes and Higson introduced the notion of asymptotic morphism of \(C^*\)-algebras. They showed that the bifunctor \(E(-,-)\), given on a pair \((A,B)\) of \(C^*\)-algebras by \[ E(A,B): =[[SA\otimes {\mathcal K},SB \otimes {\mathcal K}]], \] the group of homotopy classes of asymptotic morphisms between the stabilized suspensions of \(A\) and \(B\), is the universal stable and excisive homotopy bifunctor on the category of separable \(C^*\)-algebras. In the present article the author develops the theory further by introducing a relative version of \(E\)-theory in terms of relative asymptotic morphisms. We cite from the introduction: ``In this paper we define relative \(E\)-theory, associating to a \(C^*\)-algebra \(A\) and an ideal \(I\) the abelian groups \(E^n_{rel} (A;I)\). These groups are related to the \(E\)-theory groups of \(A\) and \(I\) in the familiar way; by a long exact sequence and excision isomorphisms. The definition of relative \(E\)-theory is motivated by the properties of first order, elliptic differential operators on complete Riemannian manifolds''.