  General relativity successfully describes space-times at scales that we can
observe and probe today, but it cannot be complete as a consequence of
singularity theorems. For a long time there have been indications that quantum
gravity will provide a more complete, non-singular extension which, however,
was difficult to verify in the absence of a quantum theory of gravity. By now
there are several candidates which show essential hints as to what a quantum
theory of gravity may look like. In particular, loop quantum gravity is a
non-perturbative formulation which is background independent, two properties
which are essential close to a classical singularity with strong fields and a
degenerate metric. In cosmological and black hole settings one can indeed see
explicitly how classical singularities are removed by quantum geometry: there
is a well-defined evolution all the way down to, and across, the smallest
scales. As for black holes, their horizon dynamics can be studied showing
characteristic modifications to the classical behavior. Conceptual and physical
issues can also be addressed in this context, providing lessons for quantum
gravity in general. Here, we conclude with some comments on the uniqueness
issue often linked to quantum gravity in some form or another.
