Precision optical astrometry of quasars and active galaxies can provide important insight into the spatial distribution and variability of emission in compact nuclei. SIM — the Space Interferometry Mission — will be the first optical interferometer capable of precision astrometry on quasars. Although it is not expected to resolve the emission, it will be very sensitive to astrometric shifts, for objects as faint as R magnitude 20. In its wide-angle mode, SIM will yield 4 microarcsecond absolute positions, and proper motions to about 2 microarcsecond/yr. A variety of AGN phenomena are expected to be visible to SIM on these scales, including time and spectral dependence in position offsets between accretion disk and jet emission. SIM should be able to answer the following questions. Does the most compact optical emission from an AGN come from an accretion disk or from a relativistic jet? Do the relative positions of the radio core and optical photocentre of quasars used for the reference frame tie change on the timescales of their photometric variability? Do the cores of galaxies harbour binary supermassive black holes remaining from galaxy mergers? In this paper we briefly describe the operation of SIM and the quasar measurements it will make. We estimate the size of the astrometric signatures which may be expected, and we discuss prospects for using astrometry as a fundamental tool for understanding quasar nuclei.