This is a report on the Oberwolfach conference “Analysis and Geometric Singularities”, May 6-12, 2012. The main themes discussed in the meeting include the interplay between the analytic, geometric and topological study of singular spaces and noncompact spaces highly regular structure at infinity and asymptotic analysis of global spectral invariants. More specific topics include index theory on such spaces, various nonlinear geometric problems, and in particular the asymptotic structure of natural metrics on geometric moduli spaces, and techniques from linear analysis to approach such problems.