The core of Herbert Edelsbrunner’s research is a combination of mathematics and computer science, always driven by relevant questions in applications. During a past shift from geometry to topology (which are related subjects without clear separation), the group observed an increase in relevant applications it could address. These include questions in scientific visualization, structural molecular biology, systems biology, but also geometry processing, medical imaging, and orthodontics. The common theme is the importance of shape and the recognition, matching, and classification of shape. Topology is the area within mathematics whose methods most directly speak to that need. Algorithms and computer software are needed to make mathematical insights useful in applications, which is the motivation to study in topology and also geometry from a computational point of view.