The conformal approach to scattering theory goes back to the 1960's
and 1980's, essentially with the works of Penrose, Lax-Phillips and
Friedlander. It is Friedlander who put together the ideas of Penrose
and Lax-Phillips and presented the first conformal scattering theory
in 1980. Later on, in the 1990's, Baez-Segal-Zhou explored Friedlander's
method and developed several conformal scattering theories. Their
constructions, just like Friedlander's, are on static spacetimes. The
idea of replacing spectral analysis by conformal geometry is however
the door open to the extension of scattering theories to general non
stationary situations, which are completely inaccessible to spectral
methods. A first work in collaboration with Lionel Mason explained
these ideas and applied them to non stationary spacetimes without
singularity. The first results for nonlinear equations on such
backgrounds was then obtained by Jeremie Joudioux. The purpose is now
to extend these theories to general black holes. A first crucial step,
recently completed, is a conformal scattering construction on
Schwarzschild's spacetime. This talk will present the history of the
ideas, the principle of the constructions and the main ingredients
that allow the extension of the results to black hole geometries.