  The spin dynamics and spin relaxation of itinerant electrons in quantum wires
with spin-orbit coupling is reviewed. We give an introduction to spin dynamics,
and review spin-orbit coupling mechanisms in semiconductors. The spin diffusion
equation with spin-orbit coupling is derived, using only intuitive, classical
random walk arguments. We give an overview of all spin relaxation mechanisms,
with particular emphasis on the motional narrowing mechanism in disordered
conductors, the D'yakonov-Perel'-Spin relaxation. Here, we discuss in
particular, the existence of persistent spin helix solutions of the spin
diffusion equation, with vanishing spin relaxation rates. We then, derive
solutions of the spin diffusion equation in quantum wires, and show that there
is an effective alignment of the spin-orbit field in wires whose width is
smaller than the spin precession length $L_{\rm SO}$. We show that the
resulting reduction in the spin relaxation rate results in a change in the sign
of the quantum corrections to the conductivity. Finally, we present recent
experimental results which confirm the decrease of the spin relaxation rate in
wires whose width is smaller than $L_{\rm SO}$: the direct optical measurement
of the spin relaxation rate, as well as transport measurements, which show a
dimensional crossover from weak antilocalization to weak localization as the
wire width is reduced. Open problems remain, in particular in narrower,
ballistic wires, were optical and transport measurements seem to find opposite
behavior of the spin relaxation rate: enhancement, suppression, respectively.
We conclude with a review of these and other open problems which still
challenge the theoretical understanding and modeling of the experimental
results.
