Abstract: When an impurity is immersed in a many-body background, it is dressed by the excitations of the bath, and forms “a polaron”. As a result, the injection spectrum of the impurity carries the hallmarks of the correlations present in the bath. This physics is relevant for excitons optically injected in a few layer heterostructure, or for cold atomic mixtures. In this talk, we will first review the basic theoretical framework and recent experimental progress. Then, we will theoretically analyze a few cases of correlated many-body states: the impurity injection spectra are predicted to display peculiar features, that allow to distinguish whether the bath features BCS pairing, charge density waves, topological phases, the BKT transition, etc.