The conventional formulations of the acoustic and electromagnetic scattering integral equations, which are valid only for surfaces with continuous and differentiable curvature, are generalized to surfaces containing curvature discontinuities. It is argued that these generalizations are essentially a generalization of the 1859 and 1882 Kirchhoff integration of the scalar and vector wave equations, respectively, as well as a generalization of the 1938 Stratton–Chu integration of Maxwell’s equations.