Notes on quantum Liouville theory and quantum gravity

There are two motivations to study two-dimensional quantum gravity. First, this theory is a toy model for four-dimensional gravity. Second, two-dimensional quantum gravity is the theory on the world-sheet of both “critical” as well as “non-critical” string theories. Some of our conclusions apply to these two more general situations. Since we would like to draw lessons from this simple theory, which are generic and valid in more complicated systems, we should study it in continuum field theory language. In these notes we review the status of the continuum Liouville approach to quantum gravity. Our understanding of the subject is far from complete. We will point out what we think are the most important open problems, and will speculate about their solutions.