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We review both the construction of conformal blocks in quantum Liouvilletheory and the quantization of Teichm\"uller spaces as developed by Kashaev,Checkov and Fock. In both cases one assigns to a Riemann surface a Hilbertspace acted on by a representation of the mapping class group. According to aconjecture of H. Verlinde, the two are equivalent. We describe some key stepsin the verification of this conjecture.Comment: Contribution to the proceedings of the 6th International Conference  on CFTs and Integrable Models, Chernogolovka, Russia, September 2002; v2:  Typos corrected, typographical change

ON THE RELATION BETWEEN QUANTUM LIOUVILLE THEORY AND THE QUANTIZED TEICHMÜLLER SPACES