Abstract: Astrometry is that fundamental part of astronomy which allows to determine the geometric, kinematical, and dynamical properties of celestial objects, including our own Galaxy, which is assembled and shaped by gravity. The knowledge of star positions was already important at the times of Hipparchus (190-120 BC) and his predecessors, Timocharis and Aristillos. Their cataloging (approximately 150 years earlier) of star positions enabled Hipparchus to update the observations with a precision of nearly half a degree and thus to discover the phenomenon of equinox precession. Nowadays a big jump is mandatory: positions, motions, and distances exist in the realm of the Einstein Theory and null geodesics represent our unique physical links to the stars through a curved space-time, namely a varying background geometry. Astrometry must be equipped with all of the proper tools of General Relativity to define the observables and the measurements needed for compiling astronomical catalogs at the microarcosecond accuracy and beyond. The astrometry of the 21st century, endowed with a fully relativistic framework, is fully fledged for new potential applications in astrophysics, can lead the way to forefront discoveries in fundamental physics, and is becoming the pillar of Local Cosmology. In this respect, it is more appropriate, in the 21st century, to refer to it as "Gravitational Astrometry".