  We systematically investigate the problem of representing a Grothendieck
topos with topological groupoids. We characterise, in model theoretic terms,
which open topological groupoids may represent the classifying topos of a
theory. Intuitively, this characterises which groupoids of models contain
enough information to reconstruct the theory. Our treatment subsumes many
previous examples of representing groupoids found in the literature. In
addition, we demonstrate that a representing groupoid for a theory remains
representing when new isomorphisms are added to the groupoid, yielding a
topological parallel to the extant theory of \'etale completeness for localic
groupoids.
