In the electromagnetic spectrum, microwaves are the waves whose wavelengths are comparable to ordinary laboratory dimensions. Furthermore, smooth surfaces of good conductors form very perfect reflectors for them. As a consequence, an electromagnetic wave in a reflecting pipe is reflected back and forth from wall to wall, so that it can travel to large distances with small attenuation. Hence we have “waveguides” as transmission lines, dealing with electromagnetic fields in the inside of hollow regions, rather than outside wires for more conventional electric applications, or waves in free space as in optics, and the propagating electromagnetic field is confined to the finite region of the guide by reflecting walls. The present article reviews the basic theory and the different geometries used in the various applications of such hollow waveguides. It also reviews specific recent (as of 2003) developments, such as theoretical advances, waveguides with arbitrary cross section, and ridged and grooved waveguides.