For the Klein–Gordon equation one formulates a characteristic Cauchy problem by specifying as initial datum the restriction to a lightcone. I will demonstrate how such characteristic Cauchy problem (also called Goursat problem) can be solved in the inside of a cone in a globally hyperbolic spacetime for data in adapted Sobolev spaces. I will then review applications in Quantum Field Theory, where one is interested in constructing solutions with a specific wave front set. (This is joint work with Christian Gérard)