In this talk, we discuss the Tannaka duality construction for the ordinary representations of a proper Lie groupoid on vector bundles. We show that for each proper Lie groupoid $G$, the canonical homomorphism of $G$ into the reconstructed groupoid $T (G)$ is surjective, although -contrary to what happens in the case of groups- it may fail to be an isomorphism. We obtain necessary and sufficient conditions in order that $G$ may be isomorphic to $T (G)$ and, more generally, in order that $T (G)$ may be a Lie groupoid. We show that if $T (G)$ is a Lie groupoid, the canonical homomorphism $G \to T(G)$ is a submersion and the two groupoids have isomorphic categories of representations.