A brief overview of universal equations, which govern the propagation of nonlinear waves in various physical media (both classical - such as optical fibers - and macroscopic quantum media, such as Bose-Einsten condensates, BECs), will be given. The survey will include the nonlinear Schroedinger (NLS)/Gross-Pitaevskii equations, the sine-Gordon (SG) equation, the Korteweg - de Vries (KdV) equation, the Kadomtsev - Petviashvili (KP) equations, and some others (the KP equations are two-dimensional models). All the above-mentioned equations, in their ideal form, share the fascinating property of the exact integrability, by means of a mathematical technique known as "the inverse-scattering transform". Not only are these equations ubiquitous and universal, but also their fundamental solutions - first of all, solitons, i.e., solitary waves - play a profoundly important role in all physical settings to which the model equations apply. Solitons have been predicted in a great variety of physical systems, and one-dimensional solitons were observed and/or created experimentally in nonlinear optics, BEC, long Josephson junctions (superconductivity), fluid flows, plasmas, etc. A great challenge to the experiment is to produce stable two- and three-dimensional solitons.