I will discuss the results of two projects using gravitational microlensing to constrain the demographics of black holes in the Milky Way, both in the stellar-mass and intermediate-mass regimes. These projects use the little-used effect of astrometric microlensing, which leads to small shifts in the apparent position of background stars as a massive lens becomes aligned with them. Using new and archival data from the Hubble Space Telescope, we are attempting to find astrometric signals of at the ~1 mas level that might be caused by black hole lenses. Because detecting astrometric microlensing signals allows for direct mass meausrements of single objects, this is currently the only method that enables us to probe the large population of single, isolated black holes that is expected to exist in the Milky Way. For the same reason, an astrometric microlensing detection would allow us to determine the mass of an intermediate-mass black hole (IMBH) without any assumptions as to the nature of the lens, making this a promising method to make the first unambiguous detection of an IMBH.