The existence of spacetime singularities is arguably one of the most disquieting predictions of general relativity. For this reason, the prevailing attitude is that some yet-to-be-discovered ‘new physics’ beyond general relativity will prevent their formation. So far, however, there is little indication as to what such new physics might be. In this seminar, I wish to present a theory-agnostic approach whereby phenomenological models describing non-singular alternatives to GR black holes can be built without committing to any specific beyond-GR framework. The phenomenology of such (GR) black hole ‘mimickers’ is rich and remarkably consequential for assessing the physical viability of these objects. I will discuss some applications, focusing in particular on stability issues. Finally, I will close with some reflections on the implications this approach might have for the search of new physics beyond GR.