Observational tests of strong field gravity are rapidly improving. This allows us to test whether the compact objects observed in the sky are truly black holes as described in general relativity or some other "exotic" objects. These tests, however, require exquisite theoretical modeling of black holes and their alternatives. This talk will give an overview of various such theoretical advancements: at the level of improved waveform modeling, as well as putting black hole mimickers under a microscope and testing their stability.