Observing Supermassive Black Holes across cosmic time: from phenomenology to physics

In the last decade, a combination of high sensitivity, high spatial resolution observations and of coordinated multi-wavelength surveys has revolutionized our view of extra-galactic black hole (BH) astrophysics. We now know that supermassive black holes reside in the nuclei of almost every galaxy, grow over cosmological times by accreting matter, interact and merge with each other, and in the process liberate enormous amounts of energy that influence dramatically the evolution of the surrounding gas and stars, providing a powerful self-regulatory mechanism for galaxy formation. The different energetic phenomena associated to growing black holes and Active Galactic Nuclei (AGN), their cosmological evolution and the observational techniques used to unveil them, are the subject of this chapter. In particular, I will focus my attention on the connection between the theory of high-energy astrophysical processes giving rise to the observed emission in AGN, the observable imprints they leave at different wavelengths, and the methods used to uncover them in a statistically robust way. I will show how such a combined effort of theorists and observers have led us to unveil most of the SMBH growth over a large fraction of the age of the Universe, but that nagging uncertainties remain, preventing us from fully understating the exact role of black holes in the complex process of galaxy and large-scale structure formation, assembly and evolution.