Equivariant representable K-theory

We interpret certain equivariant Kasparov groups as equivariant representable K-theory groups and compute these via a classifying space and as K-theory groups of suitableσ-C∗-algebras. We also relate equivariant vector bundles to theseσ-C∗-algebras and provide sufficient conditions for equivariant vector bundles to generate representable K-theory. We mostly work in the generality of locally compact groupoids with Haar systems.