We review the history of the road to a manifestly covariant perturbative calculus within quantum electrodynamics from the early semi-classical results of the mid-twenties to the complete formalism of Stueckelberg in 1934. We choose as our case study the calculation of the cross-section of the Compton effect. We analyse Stueckelberg's paper extensively. This is our first contribution to a study of his fundamental contributions to the theoretical physics of the twentieth century.