  Let $(G,\alpha)$ and $(H,\beta)$ be locally compact groupoids with Haar
systems. We define a topological correspondence from $(G,\alpha)$ to
$(H,\beta)$ to be a $G$-$H$-bispace $X$ on which $H$ acts properly and $X$
carries a continuous family of measures which is $H$-invariant and each measure
in the family is $G$-quasi invariant. We show that a topological correspondence
produces a $C^*$-correspondence from $C^*(G,\alpha)$ to $C^*(H,\beta)$. We give
many examples of topological correspondences.
