We will discuss the bubbling and neck analysis for degenerating sequences of minimal hypersurfaces which, in particular, lead to qualitative relationships between the variational, topological and geometric properties of these objects. Our aim is to discuss the salient technical details appearing in both the closed and free-boundary setting, and to give an overview of the applications of such results. This will involve expositions of joint works with Lucas Ambrozio, Reto Buzano and Alessandro Carlotto.