  Over the past five years, there has been significant progress on the problem
of quantization of diffeomorphism covariant field theories with {\it local}
degrees of freedom. The absence of a background space-time metric in these
theories gives rise to a host of conceptual and technical difficulties because
most of the familiar methods from axiomatic, constructive and perturbative
quantum field theory are no longer applicable. Perhaps the most striking
examples of these problems arise in the construction of a quantum field theory
of geometry. We show that these problems can be tackled using new
non-perturbative methods. In particular, one can rigorously define certain
geometric operators and show that their spectrum is discrete. Thus, there is a
precise sense in which the geometry is quantized at the Planck scale and the
continuum picture is only a coarse-grained approximation.
