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Testing General Relativity in the Strong-Field Regime with Observations of Black Holes in the Electromagnetic Spectrum

General relativity has been tested by many experiments, which, however, almost exclusively probe weak spacetime curvatures. In this thesis, I create two frameworks for testing general relativity in the strong-field regime with observations of black holes in the electromagnetic spectrum using current or near-future instruments. In the first part of this thesis, I design tests of the no-hair theorem, which uniquely characterizes the nature of black holes in general relativity in terms of their masses and spins and which states that these compact objects are described by the Kerr metric. I investigate a quasi-Kerr metric and construct a Kerr-like spacetime, both of which contain an independent parameter in addition to mass and spin. If the no-hair theorem is correct, then any deviation from the Kerr metric has to be zero. I show that already moderate changes of the deviation parameters in either metric lead to significant modifications of the observed signals. I apply this framework to the imaging of supermassive black holes using very-long baseline interferometry as well as to the quasi-periodic variability and relativistically broadened iron lines observed in both galactic and supermassive black holes. In the second part of this thesis, I devise a method to test the predicted evaporation of black holes in Randall-Sundrum-type braneworld gravity through the orbital evolution of black-hole X-ray binaries and obtain constraints on the size of the extra dimension from A0620-00 and XTE J1118+480.