"The project aims at deepening the relationship between various approaches to quantum gravity and noncommutative geometry.Indeed there is a strong theoretical evidence that the usual continuum model of space-time is not correct at very short distances where both gravitational and quantum mechanics effects come into play. Several theories propose a (incomplete) treatment of quantum gravity, but even the most popular of them (string theory, loop quantum gravity) have not been experimentally tested for the energy scale corresponding to the Planck length is far beyond our experimental possibilities. However it has been stressed that quantum gravity theories, by giving a non continuum structure to space-time, may lead to some observational effects. Most of the time this non continuum structure has a mathematical translation in terms of quantum, or non commutative, spacetimes. Namely one posits a noncommutative algebraic structure of the coordinates of spacetime by a deformation of the classical algebra, just as quantum groups are deformation of classical Lie algebras.However the physical status of such space-times is not clear. Naive questions like ""what are the points of a quantum space-time ?"", ""what happens to the notion of distance ?"" have no clear answer.These questions are addressed by the conceptual approach to Noncommutative Geometry (NCG) developed by Connes. NCG provides a formulation of standard geometric and topological concepts (like spin and differential structures) within a purely algebraic framework. Riemannian geometry is encompassed as a particular case, commutative, of a more general theory.The project aims at deepening the relation between the more down-to-earth-approach to quantum space-time developped by theoretical physicist on the one side, and the more fundamental (but difficult) theory of Connes on the other side.It is a continuation of EIF fellowship made by Martinetti in Amelino-Camelia`s group in Rome in 2006/08."