Accreting black holes (BHs) are believed to be sites of possible particle acceleration with favorable conditions also for effective gamma-ray production. However, because of photon-photon pair production, only low energy (MeV) gamma-rays can escape these compact objects with typically very large compactness parameter, \kappa= \fracLL_ Edd \fracR_gR >= 0.01, given that in most cases the accretion disks within 10 Schwarzschild radii R_g radiate with a power exceeding 10 percent of the Eddington luminosity, L_Edd. Therefore the high energy gamma-ray emission of these objects (both of stellar mass and super-massive BHs) is generally suppressed, and consequently the unique information on possible particle acceleration processes near the event horizon of the BH is essentially lost. Fortunately this is not the case for the super-massive BH located at the dynamical center of our Galaxy (Sgr A*), which thanks to its extraordinary low bolometric luminosity ( <= 10-8 L_Edd) is transparent for gamma-rays up to very high energies, E 10 TeV. We discuss different scenarios of gamma-ray production in Sgr A*, and show that for a reasonable set of parameters one can expect detectable gamma-ray fluxes of both hadronic and electronic origin. Some of these scenarios are applicable not only for the TeV gamma-ray emission recently reported from the direction of Galactic Center, but may have broader implications relevant to highly variable nonthermal emission of Sgr A* in radio, IR and X-ray bands.