The theory of the exponential transmission line is developed. It is found to be a high pass, impedance transforming filter. The cutoff frequency depends upon the rate of taper. The deviation of the exponential line from an ideal impedance transformer may be decreased by an order of magnitude by shunting the low impedance end with an inductance and inserting a capacitance in series with the high impedance end. The magnitudes of these reactances are equal to the impedance level at their respective ends of the line at the cutoff frequency. For a two-to-one impedance transformer the line is 0.0551 wave-lengths long at the cutoff frequency. For a four-to-one impedance transformer the line is 0.1102 wave-lengths long at the cutoff frequency, etc. The results have been verified experimentally. Practical lines 50 meters and 15 meters long have been constructed which transform from 600 to 300 ohms over the frequency range from 4 to 30 mc. with deviations from the ideal that are small compared with the deviations from the ideal of commercial transmission lines, either two-wire or concentric. When an exponential line is used as a dissipative load of known impedance instead of a uniform line it is possible to approach more nearly the ideal of constant heat dissipation per unit length. This makes it possible to use a shorter line.