We examine the scaling of geodesic correlation functions in two-dimensional
gravity and in spin systems coupled to gravity. The numerical data support the
scaling hypothesis and indicate that the quantum geometry develops a
non-perturbative length scale. The existence of this length scale allows us to
extract a Hausdorff dimension. In the case of pure gravity we find d_H approx.
3.8, in support of recent theoretical calculations that d_H = 4. We also
discuss the back-reaction of matter on the geometry.Comment: 16 pages, LaTeX format, 8 eps figure