Twists over étale groupoids and twisted vector bundles

Inspired by recent papers on twisted K-theory, we consider in this article the question of when a twist R over a locally compact Hausdorff groupoid G (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of G. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of G. When G is an étale groupoid, we establish conditions (involving the classifying space BG of G) which imply that a torsion twist R over G admits a twisted vector bundle.