  General relativity has been widely tested in weak gravitational fields but
still stands largely untested in the strong-field regime. According to the
no-hair theorem, black holes in general relativity depend only on their masses
and spins and are described by the Kerr metric. Mass and spin are the first two
multipole moments of the Kerr spacetime and completely determine all
higher-order moments. The no-hair theorem and, hence, general relativity can be
tested by measuring potential deviations from the Kerr metric affecting such
higher-order moments. Sagittarius A* (Sgr A*) is a prime target for precision
tests of general relativity with several experiments across the electromagnetic
spectrum. First, near-infrared (NIR) monitoring of stars orbiting around Sgr A*
with current and new instruments is expected to resolve their orbital
precessions. Second, timing observations of radio pulsars near the Galactic
center may detect characteristic residuals induced by the spin and quadrupole
moment of Sgr A*. Third, the Event Horizon Telescope, a global network of mm
and sub-mm telescopes, aims to study Sgr A* on horizon scales and to image its
shadow cast against the surrounding accretion flow using very-long baseline
interferometric (VLBI) techniques. Both NIR and VLBI observations may also
detect quasiperiodic variability of the emission from the accretion flow of Sgr
A*. In this review, I discuss our current understanding of the spacetime of Sgr
A* and the prospects of NIR, timing, and VLBI observations to test its Kerr
nature in the near future. [abridged]
