Quantum gravity with a positive cosmological constant

A quantum theory of gravity is described in the case of a positive cosmological constant in 3+1 dimensions. Both old and new results are described, which support the case that loop quantum gravity provides a satisfactory quantum theory of gravity. These include the existence of a ground state, discoverd by Kodama, which both is an exact solution to the constraints of quantum gravity and has a semiclassical limit which is deSitter spacetime. The long wavelength excitations of this state are studied and are shown to reproduce both gravitons and, when matter is included, quantum field theory on deSitter spacetime. Furthermore, one may derive directly from the Wheeler-deWitt equation, Planck scale, computable corrections to the energy-momentum relations for matter fields. This may lead in the next few years to experimental tests of the theory. To study the excitations of the Kodama state exactly requires the use of the spin network representation, which is quantum deformed due to the cosmological constant. The theory may be developed within a single horizon, and the boundary states described exactly in terms of a boundary Chern-Simons theory. The Bekenstein bound is recovered and the N bound of Banks is given a background independent explanation. The paper is written as an introduction to loop quantum gravity, requiring no prior knowledge of the subject. The deep relationship between quantum gravity and topological field theory is stressed throughout.