The D.C. and the A.C. conductivities of both amorphous germanium and amorphous silicon, prepared by thermal evaporation and R.F. sputtering in argon or argon-hydrogen atmospheres have been investigated. D.C. conductivity measurements were performed in the temperature range between 1.3K and 300K, up to applied electric fields of 107 V/m. The conductivity parameters are obtained for all the samples and tabulated as a function of preparation and annealing conditions and hydrogenation. The temperature dependence has been found to obey the T-1/4 hopping law in our temperature range, giving To values insensitive to the preparation conditions. The analysis of the high field dependence of the D.C. conductivity by the Apsley and Hughes (1974, 1975, 1976) theory, however, shows that the tunnelling exponent alpha-1 is a strongly increasing function of impurities and hydrogenation and when independently obtained alpha-1 values are used in the determination of the density of states, N(EF) it has been shown that N(EF) is a decreasing function of increasing impurity content or hydrogenation. A.C. conductivity measurements were performed over a similar temperature range at audio frequencies. The frequency dependence of the A.C. conductivity, above a low frequency cut-off has a power law region with S, a decreasing function of temperature. The low frequency cut-off conductivity has been observed to be thermally activated with an activation energy that is a function of the method of preparation. The interpretation of the results in terms of the-conventional tunnelling models (e.g. Austin and Mott (1969)) does not yield a satisfactory agreement. A phenomenological model based on the classical barrier hopping theory (Pike 1972) has been developed and used to explain the A.C. results. The observed cut-off in the A.C. conductivity is attributed to an abrupt cut-off in the spatial distribution of the contributing states, and the magnitudes of the cut-off activation energies suggest that this distribution of states is associated with the void structure in the samples. The D.C. field dependence of the A.C. conductivity was also observed up to D.C. fields of 107 V/m and the results were satisfactorily explained by the new model. The consistency of the changes in the A.C. model parameters with sample preparation conditions, annealing and hydrogenation were excellent. The A.C. parameters were also found to be correlated with the D.C. conductivity and this is successfully explained using the model.