  We define a bicategory with \'etale, locally compact groupoids as objects and
suitable correspondences, that is, spaces with two commuting actions as arrows;
the 2-arrows are injective, equivariant continuous maps. We prove that the
usual recipe for composition makes this a bicategory, carefully treating also
non-Hausdorff groupoids and correspondences. We extend the groupoid C*-algebra
construction to a homomorphism from this bicategory to that of C*-algebra
correspondences. We describe the C*-algebras of self-similar groups,
higher-rank graphs, and discrete Conduch\'e fibrations in our setup.
