  Random walks of particles on a lattice are a classical paradigm for the
microscopic mechanism underlying diffusive processes. In deterministic walks,
the role of space and time can be reversed, and the microscopic dynamics can
produce quite different types of behavior such as directed propagation and
organization, which appears to be generic behaviors encountered in an important
class of systems. The various aspects of classical and not so classical walks
on latices are reviewed with emphasis on the physical phenomena that can be
treated through a lattice dynamics approach.
