  The problem of finding the quantum theory of the gravitational field, and
thus understanding what is quantum spacetime, is still open. One of the most
active of the current approaches is loop quantum gravity. Loop quantum gravity
is a mathematically well-defined, non-perturbative and background independent
quantization of general relativity, with its conventional matter couplings. The
research in loop quantum gravity forms today a vast area, ranging from
mathematical foundations to physical applications. Among the most significative
results obtained are: (i) The computation of the physical spectra of
geometrical quantities such as area and volume; which yields quantitative
predictions on Planck-scale physics. (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. This discreteness emerges naturally
from the quantum theory and provides a mathematically well-defined realization
of Wheeler's intuition of a spacetime ``foam''. Long standing open problems
within the approach (lack of a scalar product, overcompleteness of the loop
basis, implementation of reality conditions) have been fully solved. The weak
part of the approach is the treatment of the dynamics: at present there exist
several proposals, which are intensely debated. Here, I provide a general
overview of ideas, techniques, results and open problems of this candidate
theory of quantum gravity, and a guide to the relevant literature.
