  We use correspondences to define a purely topological equivariant bivariant
K-theory for spaces with a proper groupoid action. Our notion of correspondence
differs slightly from that of Connes and Skandalis. We replace smooth
K-oriented maps by a class of K-oriented normal maps, which are maps together
with a certain factorisation. Our construction does not use any special
features of equivariant K-theory. To highlight this, we construct bivariant
extensions for arbitrary equivariant multiplicative cohomology theories.
  We formulate necessary and sufficient conditions for certain duality
isomorphisms in the geometric bivariant K-theory and verify these conditions in
some cases, including smooth manifolds with a smooth cocompact action of a Lie
group. One of these duality isomorphisms reduces bivariant K-theory to K-theory
with support conditions. Since similar duality isomorphisms exist in Kasparov
theory, both bivariant K-theories agree if there is such a duality isomorphism.
