Consider a family of smooth compact connected $n$ dimensional Riemannian manifolds. What can one say about the spectral geometry of a limit of these?

This question has interested many spectral geometers; my talk focuses on conical metric degeneration in which the family converges "asymptotically conically'' to an open manifold with conical singularity. I will present spectral convergence results and discuss techniques including microlocal analysis on manifolds with corners and geometric blowup constructions. I will also summarize spectral convergence results for other geometric contexts and discuss applications and open questions.