This written version of a talk given in July 2020 at the Western Sydney Abend seminar and based on the joint work [6] gives a decomposition of the C*-algebra of a locally compact groupoid with Haar system, possibly endowed with a twist, in presence of a normal subgroupoid. The natural expression of this result uses Fell bundles over groupoids. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra.