The consistent implementation of the gravitational interaction into the quantum framework is among the most important open problems in theoretical physics. A class of approaches to such a theory of quantum gravity is the application of quantization rules to Einstein’s theory of general relativity. Among these is the canonical approach which is characterized by the presence of constraints. The original choice of canonical variables were the three-metric and its momentum. More recently, a new set of variables was suggested consisting of a connection integrated around a loop in the three-manifold and its momentum (which is the flux of the triad through two-dimensional surfaces). The resulting approach is called Loop Quantum Gravity
. From a mathematical point of view it leads much further than the old geometrodynamical approach, with one of the main results being the existence of a discrete spectrum for geometric operators such as the area operator. In their topical review, the authors give a comprehensive and up-to-date introduction into this approach on a level that is suitable for non-experts who possess some knowledge of general relativity and quantum field theory.