Quantization and the Issue of Time for Various Two-Dimensional Models of Gravity

It is shown that the models of 2D Liouville Gravity, 2D Black Holeand R2Gravity are embedded in the Katanaev-Volovich model of 2D NonEinsteinian Gravity. Different approaches to the formulation of a quantum theory for the above systems are then presented: The Dirac constraints can be solved exactly in the momentum representation, the path integral can be integrated out, and the constraint algebra can be explicitely canonically abelianized, thus allowing also for a (superficial) reduced phase space quantization. Non–trivial dynamics are obtained by means of time dependent gauges. All of these approaches lead to the same finite dimensional quantum mechanical system.