  The conductance of finite systems plays a central role in the scaling theory
of localization (Abrahams et al, 1979). Usually it is defined by the
Landauer-type formulas, which remain open the following questions: (a)
exclusion of the contact resistance in the many-channel case; (b)
correspondence of the Landauer conductance with internal properties of the
system; (c) relation with the diffusion coefficient D(\omega,q) of an infinite
system. The answers to these questions are obtained below in the framework of
two approaches: (1) self-consistent theory of localization by Vollhardt and
Woelfle, and (2) quantum mechanical analysis based on the shell model. Both
approaches lead to the same definition for the conductance of a finite system,
closely related to the Thouless definition. In the framework of the
self-consistent theory, the relations of finite-size scaling are derived and
the Gell-Mann - Low functions \beta(g) for space dimensions d=1,2,3 are
calculated. In contrast to the previous attempt by Vollhardt and Woelfle
(1982), the metallic and localized phase are considered from the same
standpoint, and the conductance of a finite system has no singularity at the
critical point. In the 2D case, the expansion of \beta(g) in 1/g coincides with
results of the \sigma-model approach on the two-loop level and depends on the
renormalization scheme in higher loops; the use of dimensional regularization
for transition to dimension d=2+\epsilon looks incompatible with the physical
essence of the problem. The obtained results are compared with numerical and
physical experiments. A situation in higher dimensions and the conditions for
observation of the localization law \sigma\propto -i\omega for conductivity are
discussed.
