  We construct a complete conformal scattering theory for finite energy Maxwell
potentials on a class of curved, asymptotically flat spacetimes with prescribed
smoothness of null infinity and a non-zero ADM mass. In order to define the
full set of scattering data, we construct a Lorenz-like gauge which makes the
field equations hyperbolic and non-singular up to null infinity, and reduces to
an intrinsically solvable ODE on null infinity. We develop a method to solve
the characteristic Cauchy problem from this scattering data based on a theorem
of H\"ormander. In the case of Minkowski space, we further investigate an
alternative formulation of the scattering theory by using the Morawetz vector
field instead of the usual timelike Killing vector field.
