Gaussian beams play such an important role in optical lasers as well as in longer wavelength systems that they have been extensively analyzed, starting with some of the classic treatments mentioned in Chapter 1. Almost every text on optical systems discusses Gaussian beam propagation in some detail, and several comprehensive review articles are available. However, for millimeter and submillimeter wavelength systems there are naturally certain aspects that deserve special attention, and we emphasize aspects ofquasioptical propagation that have proven to be of greatest importance at these relatively long wavelengths. In the following sections we first give a derivation of Gaussian beam formulas based on the paraxial wave equation, in cylindrical and in rectangular coordinates. We discuss normalization, beam truncation, and interpretation of the Gaussian beam propagation formulas. We next cover higher order modes in different coordinate systems and consider the effective size of Gaussian beam modes. We then present inverse formulas for Gaussian beam propagation, which are of considerable use in system design. Finally, we consider the paraxial approximation in more detail and present an alternative derivation of Gaussian beam propagation based on diffraction integrals.