In the last two decades, it has been witnessed an increasing Use of SF6 gas insulated sUbstations, GIS, both in America and Europe. The entirely different behavior of the new type of insulation used in GIS, in comparison with conventional air insulation necessitates a rather different safety Philosophy regarding overvoltages. Since the damage in the case of be lowér.failure is more extensive, the acCeptable risk must therefore to the strong increase in cost with increasing insulation However, overinsulation has to be avoided Owing level. Careful investigations of the properties of surges, insulation characteristics, and the use of arresters are the prerequisites for SOlving the problem of insulation Coordination. After a survey of Possible overvoltages observed in extra high Voltage (i.e., 345 KV ~ V rn ~ 765 KV) sYstems, the author concludes that the lightning surges which enter the substation are of primary concern for dimensioning of the SF6 insulation. In this the author starts from an already existing SF6 gas insulated substation modelled as a coaxial cylinder without any branches (i.e., the worst case since the surge won't be further attenuated) and connected to an overhead line situated in a locality of a certain keraunic level (i.e., the number of days in the year on which thunder is heard). At the entrance and termination of the bus duct are located respectively a lightning arrester paralleled by a capacitor and an open breaker. According to the work of Berger on Mount San Salvatore near Lugano Switzerland, about 55% of alI flashes have more than one stroke. This result led Colclaser and his associates to consider the open breaker case which is based on the fact that after a circuit breaker opens on the first stroke, the second stroke will then meet an open line with the possibility of higher overvoltages due to changes in the reflection conditions. An expression of the correlation between the flash counts to earth and the keraunic level, deduced from Prentice's comprehensive summation of various researchers, enables the author to compute the earth flash density at the overhead line location. This result, together with Whitehead's expression for the postulated electrical shadow width, permit the computation of the number of flashes intercepted by the overhead line. The shielding failure rate, SFR, which expresses the .number of flashes bypassing the ground wires and contacting the phase conductor is calculated by the use of Kostenko's equation which is based on large field data. The shielding failure mechanism is well illustrated by the author by referring to the electrogeometric theory developed in 1963 by Young, Clay ton, Hileman and recently by Whitehead and his associates. The electrogeometric model together with the keraunic level, the voltage-time characteristic of the tower insulation and the probabilistic expression about stroke amplitudes based on large field data, enable the author to evaluate the shielding failure flashover rate, SFO, which expresses the number of flashes contacting the phase conductor and collapsing the insulator of the tower. The waveshape of the first stroke of one of those non chopped flashes which travels along the phase conductor towards the substation is first linearized and then approximated by the use of updating statistics. A recent algorithm developed by Inoue was used to calculate its distortion and attenuation for three different travelled lengths. For each of the three calculated waveforms, the overvoltage stresses at the entrance and termination of the GIS and at the arrester top are calculated and plotted by the use of digital computer programs. Results are observed for variations of the parameters such as distance from the arrester to the open breaker and characteristics of the lightning arrester.