The Center of our Galaxy is a peculiar region where a number of crucial astrophysical phenomena take place, from star formation to SN explosions and accretion onto a massive black hole. The quest for a massive black hole in the Galactic Nucleus is of course of particular relevance because, it would be the closest of such extreme objects, which are now believed to reside in most of the galactic nuclei of the universe. I will review here the main observational characteristics of the Galactic Center with particular attention to the the problem of existence, physical condition and activity of the 3 106 M_o black hole coincident with the compact radio source Sgr A^*. I will report historical and recent results of high energy observations of the central degree of our Galaxy, along with the specific accretion models proposed to account for the apparent lack of high energy activity from Sgr A^*. The scientific perspectives of the next X and gamma -ray missions in the domain of the Galactic Center physics are also mentioned.