The goal of this review is to present an introduction to loop quantum gravity—a background-
independent, non-perturbative approach to the problem of unification of general relativity …

The problem of describing the quantum behavior of gravity, and thus understanding
quantum spacetime, is still open. Loop quantum gravity is a well-developed approach to this …

Quantum gravity is expected to be necessary in order to understand situations in which
classical general relativity breaks down. In particular in cosmology one has to deal with …

A new functional calculus, developed recently for a fully non-perturbative treatment of
quantum gravity, is used to begin a systematic construction of a quantum theory of geometry …

The asymptotic safety scenario in quantum gravity is reviewed, according to which a
renormalizable quantum theory of the gravitational field is feasible which reconciles …

Applications of Riemannian quantum geometry to cosmology have had notable successes.
In particular, the fundamental discreteness underlying quantum geometry has led to a …

An anomaly-free operator corresponding to the Wheeler-DeWitt constraint of Lorentzian, four-
dimensional, canonical, non-perturbative vacuum gravity is constructed in the continuum …

Analytical and numerical methods are developed to analyze the quantum nature of the big
bang in the setting of loop quantum cosmology. They enable one to explore the effects of …

A functional calculus on the space of (generalized) connections was recently introduced
without any reference to a background metric. It is used to continue the exploration of the …

Quantization of diffeomorphism invariant theories of connections is studied and the quantum
diffeomorphism constraint is solved. The space of solutions is equipped with an inner …