  These are extended lecture notes for a mini course at the Spring School on
Non-Archimedean Geometry and Eigenvarieties held at Heidelberg University in
March 2023. The goal of the course is to explain a modern take on the
eigenvariety machine in the language of adic spaces. For this we build on the
theory developed in the first week of the school. We also explain some basic
selective non-archimedean functional analysis.
