Since the first description in 1834 of the "wave of translation", currently called soliton, by John Scott Russell, scientist have studied intensely such particular solutions of nonlinear wave equations i.e., coherent structures that do not change shape as they propagate. They have been put to good use in nonlinear optics and telecommunications, and play an important role in understanding the formation of large waves in oceans and in analyzing large systems of quantum particles. Moreover their importance in describing the large time behavior of nonlinear wave models is summarized by the following: Asymptotic Completeness Conjecture: any initial data of a nonlinear wave equations evolves into a superposition of coherent structures plus a part that radiates to infinity. My presentation will summarize both our current knowledge on existence of coherent structures and recent progress towards solving the asymptotic completeness conjecture.