Three different approaches to 2d Gravity will be discussed.
The first one is the continuous approach, in which the theory is defined through the functional integral over the Riemannian metric with appropriate gauge fixing. The choice of the conformal gauge leads to quantum Liouville theory and for that reason this approach is often called the Liouville Gravity.
The second one is the discrete approach, based on the idea of approximating the fluctuating 2d geometry by an ensemble of planar graphs, so that the continuous theory is recovered in the scaling limit where the planar graphs of very large size dominate. The discrete approach is usually refereed to as the Matrix Models, since technically the ensemble of the graphs is usually generated by the perturbative expansion of the integral over $N\times N$ matrices.
The third approach is 2d topological gravity. Witten built axiomatics of this theory by studying intersection theory on the moduli space of Riemann surfaces.
It will be shown in what sense all these approaches are equivalent.