Numerical techniques for Maxwell's equations or the Helmholtz equation often encounter difficulties of accuracy and convergence with scattering structures that feature cavities and sharp edges and are of moderate or large size (in wavelengths). Conversion to an equivalent integral equation formulation usually does not ameliorate the difficulty. This paper surveys some recent progress in techniques that address these difficulties for a variety of canonical and noncanonical structures. While some canonical cavity problems admit a purely analytical approach, another promising approach employs processes of analytical regularization that transform the basic integral equations to a second kind Fredholm matrix equation. The main tool is the Abel integral transform applied to trigonometric and other appropriate functions of hypergeometric type. The transformed equations are well conditioned (in contrast to the original formulation derived from Maxwell's equations); standard numerical techniques for their solution are easily applicable; near- and far-field scattered field results may be computed reliably and accurately. The process in spherical and spheroidal geometry is illustrated for the simplest canonical problem of a cavity excited by an axially located electric dipole. Extension of the technique to arbitrarily shaped (nonsymmetric) cavities has great practical implications, and progress on two-dimensional structures is described. Finally, recent developments from another class of noncanonical scatterers, arbitrarily shaped bodies of revolution, are reviewed.