Loop Quantum Gravity

Loop quantum gravity is a non-perturbative and background independent quantization of general relativity, with its conventional matter couplings. The research in loop quantum gravity forms a vast area, ranging from mathematical foundations to physical applications. Among the significative results obtained are: (i) The computation of the physical spectra of geometrical quantities such as area and volume. (ii) A derivation of the Bekenstein-Hawking black hole entropy formula. (iii) An intriguing physical picture of the microstructure of quantum physical space, characterized by a polymer-like Planck scale discreteness, which emerges naturally from the quantum theory and provides a mathematically well-defined realization of Wheeler's intuition of a spacetime ``foam''. I will summarize the main results and the present state of this approach.