  Our goal here is to present a detailed analysis connecting the anomalous
scaling properties of 2D simplicial quantum gravity to the geometry of the
moduli space M_{g,N} of genus g Riemann surfaces with N punctures. In the case
of pure gravity we prove that the scaling properties of the set of dynamical
triangulations with N vertices are directly provided by the large N asymptotics
of the Weil-Petersson volume of M_{g,N}, recently discussed by Manin and
Zograf. Such a geometrical characterization explains why dynamical
triangulations automatically take into account the anomalous scaling properties
of Liouville theory. In the case of coupling with conformal matter we briefly
argue that the anomalous scaling of the resulting discretized theory should be
related to the Gromov-Witten invariants of the moduli space M_{g,N}(X,\beta) of
stable maps from (punctured Riemann surfaces associated with) dynamical
triangulations to a manifold X parameterizing the conformal matter
configurations.
