On the Relation Between Quantum Liouville Theory and the Quantized Teichmüller Spaces

We review both the construction of conformal blocks in quantum Liouville theory and the quantization of Teichmüller spaces as developed by Kashaev, Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbert space acted on by a representation of the mapping class group. According to a conjecture of H. Verlinde, the two are equivalent. We describe some key steps in the verification of this conjecture.