  In this chapter, we discuss experiments that realize the discrete nonlinear
Schr\"odinger (DNLS) equations. The relevance of such descriptions arises from
the competition of three common features: nonlinearity, dispersion, and a
medium to large level of (periodic, quasiperiodic, or random) discreteness in
space. DNLS equations have been especially prevalent in atomic and molecular
physics in the study of Bose-Einstein condensates in optical lattices or
superlattices; and in nonlinear optics in the description of pulse propagation
in waveguide arrays and photorefractive crystals. New experiments in both
nonlinear optics and Bose-Einstein condensation provide new challenges for DNLS
models, and DNLS and related equations have also recently been used to make
important predictions in novel physical settings such as the study of composite
metamaterials and arrays of superconducting devices.
