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Abstract: In the early eighties, Connes developed his Noncommutative Geometry program, mostly to extend index theory to situations where usual tools of differential topology are not available. A typical situation is foliations whose holonomy does not necessarily preserve any transverse measure, or equivalently the orbit space of the action of the full group of diffeomorphisms of a manifold. In the end of the nineties, Connes and Moscovici worked out an equivariant index problem in these contexts, and left a conjecture about the calculation of this index in terms of characteristic classes. The aim of this talk will be to survey the history of this problem, and explain partly our recent solution to Connes-Moscovici's conjecture, focusing on the part concerning `quantization'. No prior knowledge of Noncommutative Geometry will be assumed, and part of this is joint work with Denis Perrot.