The purpose of this project is to push forward the development of asymptotic analysis in General Relativity in 4 essential directions. 1. Scattering on non stationary backgrounds; conformal scattering. Scattering theory is a precise tool of analysis of the asymptotic behaviour of fields, giving such information as asymptotic propagation speed, local decay of energy, equivalence between the field and its asymptotic profiles. It has been developed to considerable refinement using spectral analysis, but is not at present adapted to non self-adjoint cases or time-dependent situations. Our purpose is to extend the theory in both directions. First we propose to study the Klein paradox and superradiance on the Kerr metric (a phenomenon by which bosons can extract energy from the black hole) using spectral techniques. Second, we wish to develop an alternative approach to scattering based on Penrose's conformal compactification. It has already been applied successfully to non stationary situations in trivial topology. We mean to extend its range of validity to allow for the presence of black holes and give a different treatment of superradiance. 2. Quantum fields. The Hawking effect predicts the emission of a thermal radiation by a collapsing star and is in a way a quantum analogue of the famous no-hair theorem, since the thermal radiation is characterized only by the mass, charge and angular momentum of the resulting black hole. The first mathematically rigorous descriptions of this effect have been obtained recently and in some idealized situations: only free fields are considered; the proofs rely heavily on the detailed structure and high symmetry of the black hole background and on the details of the simplified collapse models considered. Our main goal is to go beyond these idealizations by considering perturbations of the metric, by taking the interior of the star into account, and -- more ambitiously still -- by going beyond the free field theory. The importance of these questions is clear in view of the expected universality of the Hawking effect. Linked to the Hawking effect is the simpler Unruh effect according to which a uniformly accelerated detector in Minkowski space-time ``sees'' a thermal radiation when coupled to a quantum field in its vacuum state. The rigorous study of the behaviour of detectors coupled to quantum fields around eternal or collapsing black holes is another goal of this project. 3. Stability. A fundamental question in relativity is the stability of universe models. This is a problem of paramount difficulty and only solved sofar for the flat empty universe. But one expects that as in the flat case, the behaviour of the full system is driven by that of the linearized one. The question of linear stability is therefore crucial and is also more tractable. It amounts to studying hyperbolic partial differential equations on a fixed background and understanding their stability and decay properties. We shall investigate the decay of fields for various values of the spin on a Kerr background or small perturbations of it. Brane cosmologies, related to the AdS/CFT correspondence conjecture, will also be considered, in view of developing scattering theories for non linear equations of quantum field theory on brane universes. 4. Inverse scattering. A scattering theory and the associated distribution of resonances encode part of the geometry of the background metric. The inverse scattering programme initiated recently aims at working out how much geometric information can be recovered from the knowledge of the scattering matrix or the distribution of resonances on a black hole spacetime. This is a physically natural question with direct applications to the interpretation of data from gravitational wave detectors, especially for resonances which are expected to be the measurable quantities. We shall consider inverse scattering at fixed energy and the inverse resonance problem for De Sitter-Kerr-Newman black holes.