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The center of the Galaxy is a prominent source in X-rays and gamma-rays. The study of its high-energy (HE) emission is crucial in understanding the physical phenomena taking place in this dense and extreme environment, where the closest supermassive black hole (SMBH) to us, Sgr A*, is lurking nearly invisible, today, in most of the energy spectrum. These phenomena are probably common to other galactic nuclei and may explain the feedback processes between nuclear regions and galaxies, so important for the overall evolution of the Universe. The Galactic center HE emission is very complex and consists of both thermal and non thermal radiation produced by compact and extended sources, surrounded by more diffuse components. All these objects and media are interacting with each other in the narrow and dense Central Molecular Zone (CMZ). Some of them also show relevant extensions towards the Galactic poles, indicating energetic outflows that seem to link the center to the recently observed large Galactic polar structures. In spite of the fundamental advances obtained in the last twenty five years with the most sensitive X-ray and gamma-ray observatories, several questions remain open to investigations. We review here the main observational results and the open issues on the high-energy diagnostics of the Galactic nuclear activity, focusing on processes that take place in the CMZ, and in particular discussing the role of the present and past SMBH activities in powering this region and possibly the whole Galaxy.