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The historic detection of gravitational waves paved the way for precision gravitational-wave astrophysics to blossom at unprecedented levels. In this talk, I will focus on extreme-mass-ratio inspirals (EMRIs) that consist of a primary supermassive compact object, and a stellar-mass secondary companion. Even though we are currently performing pertinent tests in order to characterize the "Kerrness" of astrophysical compact objects that reside in galactic cores, environmental effects, quantum corrections and generic multipolar deformations can break the integrability of geodesics around non-Kerr primaries and subsequently affect the gravitational-wave emission from these binaries. I will discuss the rich orbital phenomenology of EMRIs that include a supermassive rotating boson star primary, as well as potential observables from transient resonances. Finally, I will discuss the impact of resonance-crossing secondaries on the gravitational radiation emitted from such binaries, in order to shed more light into the electromagnetic degeneracies that currently exist between black holes and a plethora of exotic compact objects.