The dynamical theory of x-ray diffraction in crystals is a theory of wave propagation in periodic structures and, as such, its application to the diffraction by artificial periodic multilayered structures should, in principle, be straightforward. In practice, this is not so; the conventional theory involves a number of approximations that may fail for multilayered structures. The central problem is that of calculating multiple scattering effects: those that occur within a single bi-layer and those that involve different bi-layers. A modified Darwin dynamical theory of diffraction is formulated, which, once the scattering from a single bi-layer is known, allows the calculation of the diffraction by the full multilayer to be performed exactly without any further approximations. In particular, many-beam effects are included in a very straightforward way. For the calculation of the transmission and reflection coefficients of the single diffuse bi-layer two alternatives are offered: either one uses an improved kinematical theory (including corrections for index of refraction, absorption and total reflection) which has the advantage of rather general applicability provided the bi-layer is not too thick, or, one adopts for the single interface the special Epstein profile which allows for full consideration of multiple scattering effects. We find that the scattering by a diffuse boundary differs from that of a sharp boundary by a factor which is not of the Debye-Waller form, that there is an increased sensitivity to diffuse structure close to weak (nearly forbidden) reflections, and that the analytical treatment described here offers a very appreciable increase in numerical efficiency over recursive methods of calculation.