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We construct, using the quantum dilogarithm, a series of *-representations ofquantized cluster varieties. This includes a construction of infinitedimensional unitary projective representations of their discrete symmetrygroups - the cluster modular groups. The examples of the latter include theclassical mapping class groups of punctured surfaces.  One of applications is quantization of higher Teichmuller spaces.  The constructed unitary representations can be viewed as analogs of the Weilrepresentation. In both cases representations are given by integral operators.Their kernels in our case are the quantum dilogarithms.  We introduce the symplectic/quantum double of cluster varieties and relatedthem to the representations.Comment: Dedicated to David Kazhdan for his 60th birthday. The final version.  To appear in Inventiones Math. The last Section of the previous versions was  removed, and will become a separate pape

The quantum dilogarithm and representations of quantum cluster varieties