  We review the latest progress and properties of the families of bright and
dark one-dimensional periodic waves propagating in saturable Kerr-type and
quadratic nonlinear media. We show how saturation of the nonlinear response
results in appearance of stability (instability) bands in focusing (defocusing)
medium, which is in sharp contrast with the properties of periodic waves in
Kerr media. One of the key results discovered is the stabilization of
multicolor periodic waves in quadratic media. In particular, dark-type waves
are shown to be metastable, while bright-type waves are completely stable in a
broad range of energy flows and material parameters. This yields the first
known example of completely stable periodic wave patterns propagating in
conservative uniform media supporting bright solitons. Such results open the
way to the experimental observation of the corresponding self-sustained
periodic wave patterns.
