The fundamental structure of matter and spacetime at the shortest length scales remains an exciting frontier of basic research in theoretical physics. A unifying theme in this area is the quantization of geometrical objects. The majority of lectures at the Advanced Study Institute on Quantum Ge- ometry in Akureyri was on recent advances in superstring theory, which is the leading candidate for a unified description of all known elementary par- ticles and interactions. The geometric concept of one-dimensional extended objects, or strings, has always been at the core of superstring theory but...