  This review describes in detail the essential techniques used in microscopic
theories on spintronics. We have investigated the domain wall dynamics induced
by electric current based on the $s$-$d$ exchange model. The domain wall is
treated as rigid and planar and is described by two collective coordinates: the
position and angle of wall magnetization. The effect of conduction electrons on
the domain wall dynamics is calculated in the case of slowly varying spin
structure (close to the adiabatic limit) by use of a gauge transformation. The
spin-transfer torque and force on the wall are expressed by Feynman diagrams
and calculated systematically using non-equilibrium Green's functions, treating
electrons fully quantum mechanically. The wall dynamics is discussed based on
two coupled equations of motion derived for two collective coordinates. The
force is related to electron transport properties, resistivity, and the Hall
effect. Effect of conduction electron spin relaxation on the torque and wall
dynamics is also studied.
