  This is a review of what is currently known about the gluon Sivers
distribution and of what are the opportunities to learn more about it. Because
single transverse spin asymmetries in $p^\uparrow \, p \to \pi \, X$ provide
only indirect information about the gluon Sivers function through the relation
with the quark-gluon and tri-gluon Qiu-Sterman functions, current data from
hadronic collisions at RHIC have not yet been translated into a solid
constraint on the gluon Sivers function. SIDIS data, including the COMPASS
deuteron data, allow for a gluon Sivers contribution that is of the natural
size expected from large $N_c$ arguments, which is ${\cal O}(1/N_c)$ times the
nonsinglet quark Sivers contribution. Several very promising processes to
measure the gluon Sivers effect directly have been put forward, which apart
from ongoing and future investigations at RHIC, would strongly favor
experiments at AFTER@LHC and a possible future Electron-Ion Collider. Due to
the inherent process dependence of TMDs, the gluon Sivers TMD probed in the
various processes are not necessarily the same, but rather can be different
linear combinations of two universal gluon Sivers functions that have different
behavior under charge conjugation and that therefore satisfy different
theoretical constraints. For this reason both hadronic and DIS type of
collisions are essential in the study of the role of gluons in transversely
polarized protons.
