  Let X be a locally compact space, and let A and B be Co(X)-algebras. We
define the notion of an asymptotic Co(X)-morphism from A to B and construct
representable E-theory groups RE(X;A,B). These are the universal groups on the
category of separable Co(X)-algebras that are Co(X)-stable,
Co(X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups
are naturally isomorphic to Kasparov's representable KK-theory groups
RKK(X;A,B). Applications and examples are also discussed.
