We analyze 2-dimensional quantum gravity. By the use of light-cone gauge, we evaluate energy-momentum tenor and equation of motion. We also show that Ward identity expresses SL(2,R) Kac-Moody algebra and find differential equations for the correlation functions. To renormalize this theory we couple it to conformal field theory and find formulate for the spectrum of anomalous dimensions. In the conformal gauge quantization of gravity coupled to conformal field gives Liouville field action and shows complete agreement with the results in the light-cone gauge. We also prove that two dimensional quantum gravity connects with the Wess-Zumino-Witten model.