The mode-matching applications to scattering from circular and annular apertures in a thick perfectly conducting plane are reviewed. The Hankel and Weber transforms are utilized to solve the boundary-value problems of circular and annular apertures. Simple electrostatic problems are presented to illustrate the mode-matching method in terms of the Hankel and Weber transforms. Various types of Weber transform are discussed with boundary-value problems. Electromagnetic radiation and scattering from circular and annular aperture geometries are summarized. The utility of the mode-matching method in circular and annular aperture scattering is emphasized.