  We present a self-contained analysis of theories of discrete 2D gravity
coupled to matter, using geometric methods to derive equations for generating
functions in terms of free (noncommuting) variables. For the class of discrete
gravity theories which correspond to matrix models, our method is a
generalization of the technique of Schwinger-Dyson equations and is closely
related to recent work describing the master field in terms of noncommuting
variables; the important differences are that we derive a single equation for
the generating function using purely graphical arguments, and that the approach
is applicable to a broader class of theories than those described by matrix
models. Several example applications are given here, including theories of
gravity coupled to a single Ising spin ($c = 1/2$), multiple Ising spins ($c =
k/2$), a general class of two-matrix models which includes the Ising theory and
its dual, the three-state Potts model, and a dually weighted graph model which
does not admit a simple description in terms of matrix models.
