  Loop quantum gravity is a mature theory. To proceed to explicit calculations
in cosmology, it is necessary to make assumptions and simplifications based on
the symmetries of the cosmological setting. Symmetry reduction is especially
critical when dealing with cosmological perturbations. The present article
reviews several approaches to the problem of building a consistent formalism
that describes the dynamics of perturbations on a quantum spacetime and tries
to address their respective strengths and weaknesses. We also review the main
open issues in loop quantum cosmology.
