The textbook problem of an impurity coupled to a Fermi sea (''Fermi polaron'') has recently attracted much interest due to experiments on both ultracold atoms and semiconductors coupled to light. For strong coupling, the latter also give rise to mixed light-matter excitations (exciton-polaritons). Previous works on these systems ignored a crucial ingredient, namely that they are dynamic versions of the classic Fermi-edge singularity problem. In this talk, I will present our results [1,2] for the spectra of these systems, which are based on a new diagramatic resummation scheme we developed. Our starting point is the exactly solvable limit of an immobile impurity with infinite mass, where spectral response functions show universal power laws related to the Fermi-edge singularity. We show how these spectra are modified for mobile impurities, taking into account the appearance of a bound state. For two-body spectra, we uncover an intriguing dichotomy: while a finite mass rounds off the spectrum when the Fermi energy is large compared to the binding energy, in the opposite limit the spectrum is actually sharpened, and a quasi-particle peak reemerges.