  General relativity can be recast as a theory of connections by performing a
canonical transformation on its phase space. In this form, its (kinematical)
structure is closely related to that of Yang-Mills theory and topological field
theories. Over the past few years, a variety of techniques have been developed
to quantize all these theories non-perturbatively. These developments are
summarized with special emphasis on loop space methods and their applications
to quantum gravity.
