  No theory of four-dimensional quantum gravity exists as yet. In this
situation the two-dimensional theory, which can be analyzed by conventional
field-theoretical methods, can serve as a toy model for studying some aspects
of quantum gravity. It represents one of the rare settings in a
quantum-gravitational context where one can calculate quantities truly
independent of any background geometry. We review recent progress in our
understanding of 2d quantum gravity, and in particular the relation between the
Euclidean and Lorentzian sectors of the quantum theory. We show that
conventional 2d Euclidean quantum gravity can be obtained from Lorentzian
quantum gravity by an analytic continuation only if we allow for spatial
topology changes in the latter. Once this is done, one obtains a theory of
quantum gravity where space-time is fractal: the intrinsic Hausdorff dimension
of usual 2d Euclidean quantum gravity is four, and not two. However, certain
aspects of quantum space-time remain two-dimensional, exemplified by the fact
that its so-called spectral dimension is equal to two.
