It is argued that the capacitance of a fractal metallic cluster scales with its size as C approximately r^c, where 0<c<1 is a new charging exponent. A new dimension d_c is defined to characterise the electrostatically unshielded surface of the cluster. In effect, c=2-d+d_c is obtained, where d is Euclidean dimension. Based upon this, and using cluster-size distribution from percolation theory, it is shown that the temperature dependence of the variable-range-hopping conductivity of discontinuous metal films is given by ln sigma ^{varies as}-1/T^x with x=1/(1+c). The value of x is predicted to lie in the range ¹/₂-1, in close agreement with experiments. It is suggested that the cermet conductivities, which have x=¹/₂, can also be explained within the framework of the present arguments.