Zappa-Sz\'ep product groupoids and C*-blends

We study the external and internal Zappa-Sz'ep product of topological groupoids. We show that under natural continuity assumptions the Zappa-Sz'ep product groupoid is 'etale if and only if the individual groupoids are 'etale. In our main result we show that the C*-algebra of a locally compact Hausdorff 'etale Zappa-Sz'ep product groupoid is a C*-blend, in the sense of Exel, of the individual groupoid C*-algebras. We finish with some examples, including groupoids built from *-commuting endomorphisms, and skew product groupoids.