Astrometry - the precise measurement of celestial positions and motions - is entering the micro-arcsecond (μas) era at multiple wavelengths, enabling new insights on compact objects across all mass scales. Here we review how high-precision astrometry is advancing our understanding of compact objects - neutron stars (NSs) and black holes (BHs). We provide the context for high precision astrometry before discussing natal kicks and the latest results from Gaia Data Release 3 (DR3). We highlight the evidence for mass-dependent peculiar velocities of accreting binaries, and also reveal a close similarity between NSs and BHs. Next-generation surveys will find recoiling supermassive BHs (SMBHs) in galactic nuclei, exploring how gravitational-wave-induced kicks operate. Exploitation of scientific opportunities on the lunar surface could facilitate much larger collecting areas and astrometric precision in X-rays than currently feasible.