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 * ), the supermassive black hole at the center of the Milky Way, 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 the silhouette of 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. this regime, it is sufficient to employ a parameterized post-Newtonian framework within which suitable corrections to Newtonian gravity in flat space can be calculated [59]. In the strong-field regime, however, i.e., at radii r ∼ r g , which are targeted by observations of the accretion flow of Sgr A * with the EHT and NIR instruments such as GRAVITY, the parameterized post-Newtonian formalism can no longer be applied. Instead, a careful modeling of the underlying spacetime in terms of a Kerr-like metric (e.g., [60][61][62][63][64][65][66][67][68]) is required.Strong-field tests of general relativity with black holes have also been proposed using gravitationalwave observations of extreme mass-ratio inspirals (EMRIs) [61-63, 65, 69-80] and of gravitational ringdown radiation of perturbed black holes after a merger with another object [81][82][83]. See Refs. [84,85] for recent reviews. Likewise, strong-field tests of general relativity have been suggested using other electromagnetic observations of accretion flows in terms of their continuum spectra [12,68,[86][87][88][89][90][91][92][93][94][95], relativistically-broadened iron lines [93,94,[96][97][98][99][100][101][102][103], variability [101,104-109], X-ray polarization [91,95,110], jets [111], and other accretion properties [12,13,[112][113][114][115][116]. See Ref.[117] for a review.All three approaches to test general relativity with...