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The history of the soliton is discussed and two methods of obtaining soliton solutions are presented. The bilinear method is applied to the Boussinesq, shallow water wave and nonlinear Schroedinger equation. A number of exact soliton and breather solutions are obtained and the dependence of the solutions on the parameters is investigated. Some rational solutions to these equations are obtained and it is shown that the breather solutions to the nonlinear Schroedinger equation can be written as an imbricate series of rational growing-and-decaying mode solutions. Exact 1- and 2-soliton solutions to the nonlinear Schroedinger, modified Korteweg-de Vries and Sine-Gordon equations are obtained from the inverse scattering transform. Besides the standard initial conditions also the use of other initial conditions for inverse scattering is investigated. In the appendix a discussion of the extended homoclinic test technique is given.