Introduction.

Discreteness provides unique opportunities for controlling the flow of light. As in solidstate physics, optical discrete or lattice configurations are known to exhibit a succession of allowed Floquet-Bloch bands and forbidden bandgaps. In weakly coupled systems, the Floquet-Bloch states can be accurately described by local modes, and thus the tight-binding approximation or coupled-mode theory is applicable. As a result, the field dynamics become effectively discretized. In optics, arrays of evanescently coupled waveguides, photonic crystal fibers, chains of coupled microresonators, and photonic crystals are prime examples of such structures where discrete wave dynamics can be experimentally investigated. Perhaps the most exciting outcome of the interplay between discreteness and optica nonlinearity is the existence of self-localized entities better known as discrete solitons. This class of optical solitons were first suggested in the late 1980s and successfully observed in AlGaAs waveguide arrays a decade later. Since then, discrete optical solitons have been observed in many other material systems such as in optically induced lattices, in quadratic waveguide arrays, and in liquid-crystal cells. This experimental work has not only resulted in a deeper understanding of nonlinear processes in periodic environments, but it has also helped to assess the potential of these self-trapped states toward future applications. Nowadays, the field of discrete optical dynamics in nonlinear lattices is at an exciting stage of development. Even though some of the basic concepts in this area have been around for a while, much remains to be explored and discovered.