  Quantum geometry, i.e., the quantum theory of intrinsic and extrinsic spatial
geometry, is a cornerstone of loop quantum gravity. Recently, there have been
many new ideas in this field, and I will review some of them. In particular,
after a brief description of the main structures and results of quantum
geometry, I review a new description of the quantized geometry in terms of
polyhedra, new results on the volume operator, and a way to incorporate a
classical background metric into the quantum description. Finally I describe a
new type of exponentiated flux operator, and its application to Chern-Simons
theory and black holes.
