  This is an expository article that describes the spectral-theoretic aspects
in the study of the stability of self-similar blowup for nonlinear wave
equations. The linearization near a self-similar solution leads to a genuinely
nonself-adjoint operator which is difficult to analyze. The main goal of this
article is to provide an accessible account to the only known method that is
capable of providing sufficient spectral information to complete the stability
analysis. The exposition is based on a mini course given at the Summer School
on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.
