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. Research in loop quantum gravity today forms a vast area, ranging from mathematical foundations to physical applications. Among the most significant 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, over-completeness 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. The view of a relativist For a relativist, on the other hand, the idea of a fundamental description of gravity in terms of physical excitations over a background metric space sounds physically very wrong. The key lesson learned from general relativity is that there is no background metric over which physics happens (unless, of course, in approximations). The world is more complicated than that. Indeed, for a relativist, general relativity is much more than the field theory of a particular force. And I do not mean that they could be superseded: I mean that all their specific predictions could be disproved by experiments. Nature does not always share our aesthetic judgments, and the history of theoretical physics is full of enthusiasm for strange theories turned into disappointment. The arbiters in science are experiments, and not a single experimental result supports, not even very indirectly, any of the current theories that go beyond the Standard Model and general relativity . Loop quantum gravity is based on the formulation of classical general relativity, which goes under the name of”new variables”, or”Ashtekar variables”, or”connectio-dynamics” (in contrast to Wheeler’s”geometrodynamics”). In this formulation, the field variable is a self-dual connection, instead of the metric, and the canonical constraints are simpler than in the old metric formulation. The idea of using a self-dual connection as field variable and the simple constraints it yields were discovered by Amitaba Sen [ 190 ]. Abhay Ashtekar realized that in the SU (2) extended phase space a self-dual connection and a densitized triad field form a canonical pair [ 8 , 9 ] and set up the canonical formalism based on such pair, which is the Ashtekar formalism. Recent works on the loop representation are not based on the original Sen-Ashtekar connection, but on a real variant of it, whose use has been introduced into Lorentzian general relativity by Barbero [ 40 , 41 , 42 , 43 ]. 1986 Wilson loop solutions of the hamiltonian constraint Thus, diffeomorphism invariant physical states are labeled by knots. A knot represents an elementary quantum excitation of space. It is not here or there, since it is the space with respect to which here and there can be defined. A knot state is an elementary quantum of space. In this manner, loop quantum gravity ties the new notion of space and time introduced by general relativity with quantum mechanics. As I will illustrate later on, the existence of such elementary quanta of space is then made concrete by the quantization of the spectra of geometrical quantities. Problems not addressed Quantum gravity is an open problem that has been investigated for over seventy years now. Quantum states representing flat spacetime. Weaves. Discrete small scale structure of space . The s-knot states do not represent excitations of the quantum gravitational field over flat space, but rather over “no-space”, or over the gμv = 0 solution. A natural problem is then how flat space (or any other smooth geometry) might emerge from the theory. Notice that in a general rela-tivistic context the Minkowski solution does not have all the properties of the conventional field theoretical vacuum. I think that the classical limit, the quantum description of black holes, or graviton-graviton scattering, just to mention a few examples, could be addressed much more easily in the covariant picture. Second, it allows the general ideas of Hartle [ 102 ] and Isham [ 118 , 119 , 122 , 121 ] on the interpretation of generally covariant quantum theories to be applied in loop quantum gravity. This could drastically simplify the complications of the canonical way of dealing with general covariant observables [ 169 , 167 ]. More precisely, in terms of elementary excitations carrying discretized quanta of area. The dynamics is coded into the hamiltonian constraint. A well defined version of this constraint exists (see equation ( 35 )), and thus a consistent theory exists, but a proof that the classical limit of this theory is classical general relativity is still lacking. Alternative versions of the hamiltonian constraint have been proposed and are under investigation. In all these cases, the hamiltonian has the crucial properties of acting on nodes only. This implies that its action is naturally discrete and combinatorial. This fact is possibly at the roots of the finiteness of the theory.