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