Much work has already been done towards establishing techniques for tapering the cross-section of waveguides without introducing significant reflections when waves are propagated along them. In the present paper the problem is approached from the aspect of maintaining a completely undisturbed field pattern outside the guide whatever its cross-section, so that the wave impedance remains constant. Examples are given illustrating how this result may, in principle, be achieved both for the hollow metal guides of rectangular and circular cross-section and for single-wire transmission lines. On the basis proposed it is shown that the surface impedance of the guide must always have a resistive component which has nothing to do with losses but is dependent solely upon the angle of the taper and represents energy crossing the interface from the field on one side to the field on the other.