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Sky survey telescopes and powerful targeted telescopes play complementaryroles in astronomy. In order to investigate the nature and characteristics ofthe motions of very faint objects, a flexibly-pointed instrument capable ofhigh astrometric accuracy is an ideal complement to current astrometric surveysand a unique tool for precision astrophysics. Such a space-based mission willpush the frontier of precision astrometry from evidence of Earth-mass habitableworlds around the nearest stars, to distant Milky Way objects, and out to theLocal Group of galaxies. As we enter the era of the James Webb Space Telescopeand the new ground-based, adaptive-optics-enabled giant telescopes, byobtaining these high precision measurements on key objects that Gaia could notreach, a mission that focuses on high precision astrometry science canconsolidate our theoretical understanding of the local Universe, enableextrapolation of physical processes to remote redshifts, and derive a much moreconsistent picture of cosmological evolution and the likely fate of our cosmos.Already several missions have been proposed to address the science case offaint objects in motion using high precision astrometry missions: NEAT proposedfor the ESA M3 opportunity, micro-NEAT for the S1 opportunity, and Theia forthe M4 and M5 opportunities. Additional new mission configurations adapted withtechnological innovations could be envisioned to pursue accurate measurementsof these extremely small motions. The goal of this White Paper is to addressthe fundamental science questions that are at stake when we focus on themotions of faint sky objects and to briefly review instrumentation and missionprofiles.

Faint objects in motion: the new frontier of high precision astrometry