The need for reliable and accurate prediction of electromagnetic wave coupling to, or scattering from, structures arises in many contexts. General-purpose computational codes have been extensively developed in the last few decades as computing power and resources have become widely available; they have had a significant impact in providing numerical solutions and insight into important coupling and scattering mechanisms. However, their accuracy, particularly for objects of some complexity, incorporating edges and re-entrant structures, can be difficult to assess. The strongly resonant features of cavity-backed apertures can present difficulties in accuracy and computational cost for such general-purpose numerical codes. This chapter presents a method of analytically regularizing the underlying integral equation governing diffraction from the structure, so that a well-conditioned system of equations is obtained. It generalizes the process of analytical regularization applied to cavities of spherical and other canonical shape [2, 3] in which the basic equations are transformed to a second-kind Fredholm matrix equation. It applies to axisymmetric bodies, and examples confirm that the condition number of the resultant system is well controlled even near-resonant frequencies and that solutions of guaranteed accuracy can be efficiently obtained.