Abstract The accurate analysis and characterization of waveguide discontinuities is an important issue in microwave engineering. In some cases, the discontinuities are an unavoidable result of mechanical or electric transitions which effects have to be minimized; in other cases, the discontinuities are deliberately introduced into the waveguide to perform a certain electric function. This is the first of two papers analyzing different types of rectangular waveguide discontinuities by using a fully automatic hp-adaptive finite element method. In this paper, a fully automatic energy-norm based hp-adaptive Finite Element (FE) strategy applied to a number of relevant waveguide structures, is presented. The methodology produces exponential convergence rates in terms of the energy-norm error of the solution against the problem size (number of degrees of freedom). Extensive numerical results demonstrate the suitability of the hp-method for solving different rectangular waveguide discontinuities. These results illustrate the flexibility, reliability, and high-accuracy of the method. The self-adaptive hp-FEM provides similar (sometimes more) accurate results than those obtained with semi-analytical techniques such as the mode matching method, for problems where semi-analytical methods can be applied. At the same time, the hp-FEM provides the flexibility of modeling more complex waveguide structures and including the effects of dielectrics, metallic screws, round corners, etc., which cannot be easily considered when using semi-analytical techniques. In the second part, the work is extended by presenting a fully automatic goal-oriented hp-adaptive FE in terms of the scattering or S-parameters which are widely used in microwave engineering for the characterization of microwave devices.