We construct a complete conformal scattering theory for Maxwell fields in the static exterior region of a Reissner-Nordstrøm-de Sitter black bole spacetime. This is done using uniform energy decay results that we obtain in a separate paper [47], to show that the trace operators are injective and have closed ranges. We then solve the Goursat problem (characteristic Cauchy problem) for Maxwell fields on the null boundaries showing that the trace operators are also surjective.1 arXiv:1706.06993v1 [gr-qc] 21 Jun 2017 profile to gain information about the medium and the obstacles it contains. This reconstruction is the aim of inverse scattering. Analytic Scattering: Brief literature OverviewScattering theory proved to be a useful tool in the framework of general relativity to study the asymptotic influence of the geometry of spacetime on fields. Although in this current work we do not use an analytic approach to scattering, we very briefly touch on the history of the subject because this is part of the origin of conformal scattering and it helps to understand what new features the conformal approach bring to the domain. Scattering theory in black holes spacetimes played an essential role in the rigorous description of phenomena like superradiance, the Hawking effect, and quasi-normal modes (resonances of black holes which are related to gravitational waves). In 1980 S. Chandrasekhar [12] used the stationary approach, resorting to a Fourier transformation in time, to study quasi-linear modes of black hole spacetimes such as Schwarzschild, Reissner-Nordstrøm, and Kerr. Chandrasekhar's work systematically used the Newman-Penrose formalism to develop stationary scattering theories described in terms of the scattering matrix of transmission and reflection coefficients. And around the same time, M. Reed and B. Simon published "Scattering Theory" the third volume of their classic series [57]. Then time-dependent scattering (based on the comparison of dynamics) of classical and quantum fields on the exterior of a Schwarzschild black hole were first studied by J. Dimock in 1985 [24] and by J. Dimock and 1987 [25, 26, 27]. And in the 1990's, A. Bachelot produced an important series of papers starting with scattering theories for classical fields, Maxwell in 1990 and 1991 [1, 3], Klein-Gordon in 1994 [4] and on the Hawking effect for a spherical gravitational collapse in 1997 [2], 1999 [5] and 2000 [6]. J.-P. Nicolas in 1995 developed a scattering theory for classical massless Dirac fields [49], and a work on a non linear Klein-Gordon equation on the Schwarzschild metric (and other similar geometries) with partial scattering results obtained by conformal methods in 1995 [50]. W.M. Jin in 1998 contributed to the subject with a construction of wave operators in the massive case [36], and F. Melnyk in 2003 obtained a complete scattering for massive charged Dirac fields [44] and the Hawking effect for charged, massive spin-1/2 fields [45]. In 1999 I. Laba and A. Soffer [39] obtained complete scattering for the nonlinear Schrödinger...