In this report we summarize the research in mathematics and theoretical physics done in project A6 of the SFB 676. The main research directions include the study of internal spaces of string compactifications, geometry of scalar manifolds and moduli spaces, heterotic/type II string duality, partial breaking of supersymmetry and supersymmetric AdS backgrounds. Typical results are the construction of a wealth of new inhomogeneous complete quaternionic K\"ahler manifolds based on a combination of ideas from supergravity and perturbative string theory with new mathematical tools and the derivation of low energy effective actions in physically relevant situations, among other examples.