In this project I will work on several topics from convex and integral geometry. The focus will lie on the study of valuations. These are classical geometric quantities like volume and surface, which behave naturally with respect to unions and intersections of sets. In the last decades, valuations were studied in the context of convex bodies, convex functions and on curved objects resp. surfaces in higher-dimensional spaces, so-called manifolds, and multiple similarities between the different theories were discovered. The aim of this project is to develop new methods to confirm these empirical observations and to make the relations explicit. Subsequently, I will apply these methods in the integral geometry of certain manifolds (complex resp. quaternionic projective spaces) as well as to solve current problems for valuations on convex bodies resp. convex functions.