Scattering theories are of great importance for many problems in General Relativity, and scattering in the interior of a black hole is particularly relevant in the context of the cosmic censorship conjecture and the related Cauchy horizon instability problem. The Cauchy horizon instability is thought to be directly linked to a notion of gravitational blue-shift at the horizon, which manifest itself as a blow-up in some observed quantity. By constructing the scattering channels, one aims to recover information about the behavior of the field near the horizons, where the instabilities might be seen as the unboundedness of the scattering operators (energy blow-up) or from the lack of regularity at the horizon of the propagating field (C^1-blow-up). In this talk I will present the recent development of the scattering of waves and Dirac fields in the interior of spherically symmetric black holes between the Cauchy horizon and the event horizon. From a mathematical point of view, these two fields exhibit contrasting scattering phenomena. The scattering theory for Dirac fields is robust thanks to the conserved Dirac current and the well posedness of the characteristic Cauchy problem, even though some care must be exercised when the ambient charge is taken into account. On the other hand, the scattering of linear waves is very delicate and interesting (and surprising!) breakdowns of scattering happens in generic situations. The physical implications and causes for these breakdowns are not understood yet, and some of them may be related to possibly new phenomena.
