  In this review we discuss the interplay between discretization, constraint
implementation, and diffeomorphism symmetry in Loop Quantum Gravity and Spin
Foam models. To this end we review the Consistent Discretizations approach,
which is an application of the master constraint program to construct the
physical Hilbert space of the canonical theory, as well as the Perfect Actions
approach, which aims at finding a path integral measure with the correct
symmetry behavior under diffeomorphisms.

General Relativity and Quantum Cosmology;High Energy Physics - Lattice;High Energy Physics - Theory