Astrometry - the mapping of the positions and motions of astronomical objects - is in a period of rapid progress, largely due to the successful GAIA mission. Progress in both the precision of the measurements as well as the sheer number of objects are giving us new insights into the dynamics of the Milky Way, and raises the question as to how one might use this information to learn about the invisible matter in the galaxy. I will review recent progress made trying to understand how weak lensing from dark matter in the Milky Way could show up in the apparent motions of luminous objects, and how that can be used to constrain properties of dark matter, both now and in the future.