  Transport phenomena in spatially periodic systems far from thermal
equilibrium are considered. The main emphasize is put on directed transport in
so-called Brownian motors (ratchets), i.e. a dissipative dynamics in the
presence of thermal noise and some prototypical perturbation that drives the
system out of equilibrium without introducing a priori an obvious bias into one
or the other direction of motion. Symmetry conditions for the appearance (or
not) of directed current, its inversion upon variation of certain parameters,
and quantitative theoretical predictions for specific models are reviewed as
well as a wide variety of experimental realizations and biological
applications, especially the modeling of molecular motors. Extensions include
quantum mechanical and collective effects, Hamiltonian ratchets, the influence
of spatial disorder, and diffusive transport.
