In this talk we will survey the notion of a relative Calabi-Yau structure on a linear stable infinity-category and how examples of such relative Calabi-Yau structures naturally arise from categorified perverse sheaves on surfaces. Such categorified perverse sheaves are called perverse schobers. The infinity-category of global sections of a perverse schober F is also referred to as the topological Fukaya category of the surface with coefficients in F.