  General Relativity describes gravity in geometrical terms. This suggests that
quantizing such theory is the same as quantizing geometry. The subject can
therefore be called quantum geometry and one may think that mathematicians are
responsible of this subject. Unfortunately, most mathematicians are not aware
of this beautiful area of study. Here we give a basic introduction to what
quantum geometry means to a community working in a theory known as loop quantum
gravity. It is directed towards graduate or upper students of physics and
mathematics. We do it from a point of view of a mathematician.
