  We describe microscopic theory for the quantum transport through finite
interacting systems connected to noninteracting leads. It can be applied to
small systems such as quantum dots, quantum wires, atomic chain, molecule, and
so forth. The Keldysh formalism is introduced to study the nonlinear
current-voltage characteristics, and general properties of the nonequilibrium
Green's functions are provided. We apply the formulated to an
out-of-equilibrium Anderson model that has been used widely for the Kondo
effect in quantum dots. In the linear-response regime the Kubo formalism still
has an advantage, because it relates the transport coefficients directly to the
correlation functions defined with respect to the thermal equilibrium, and it
has no ambiguities about a profile of the inner electric field. We introduce a
many-body transmission coefficient, by which the dc conductance can be
expressed in a Landauer-type form, quite generally. We also discuss transport
properties of the Tomonaga-Luttinger liquid, which has also been an active
field of research in this decade.
