Loop Quantum Gravity, also called Quantum Gravity, is an attempt to unify General Relativity and Quantum Mechanics which has now been developed for twenty years. The aim of the Loop Quantum Gravity is to construct a mathematically rigorous background independent non-perturbative quantum theory for Lorentzian General Relativity on four dimensional manifolds.
In this work, we present a pedagogical introduction to Loop Quantum Gravity. We emphasize the physical basic notions and interpretations of the theory as well as the fundamental mathematical structure of the formalism. The theory is described on its two sides, the so-called loop representation and the so-called connection representation, and the relation between both representations is highlighted. We also give a sketch of the introduction of Ashtekar new variables in the frame of the ADM formalism of General Relativity.