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We first review the introduction of star products in connection withdeformations of Poisson brackets and the various cohomologies that are relatedto them. Then we concentrate on what we have called ``closed star products" andtheir relations with cyclic cohomology and index theorems. Finally we shallexplain how quantum groups, especially in their recent topological form, are inessence examples of star products.

Closedness of Star Products and Cohomologies