In this work the basic theory of circular corrugated horn antennas is presented, using easy concepts with simple mathematics. The aim is to give the reader an understanding of the fundamental electromagnetic properties of these devices and to provide a set of quantitative design tools for the electromagnetic engineer. The wave propagation problem in corrugated waveguides is approached by approximating the field boundary conditions on the corrugation limit surface and by making further assumptions, suggested by practical considerations, with a consequent mathematical simplification of the theory. This approach has the major advantage of providing an analytical solution for the electromagnetic field, which is accurate as far as the approximate boundary conditions are valid and that enables us to highlight the most relevant features of corrugated waveguides. The problem of radiation from corrugated horns is addressed in a similar way by choosing an approximated model sufficiently accurate for almost all practical needs. This model, the “Gaussian beam model,” has a simple analytical formulation and is rigorously treated. The final result of this approach is a set of analytical formulas relating electromagnetic properties to geometric parameters that are very useful in the design. Then, further design aspects concerning the throat region, the phase center, and cross-polarization are introduced and a numerical methods for highly accurate analysis of corrugated horns are described, ending with some design examples.