Over the last half-century, there has been increased interest in many integrable nonlinear evolution equations, such as the Nonlinear Schrodinger, Sine-Gordon, Korteweg-de Vries, and Benjamin-Ono equations, due to their ubiquity in describing physical systems exhibiting weakly nonlinear and dispersive behavior along with the remarkable fact that all of these equations have the same qualitative solution method, generally known as an Inverse Scattering Transform. A peculiar phenomenon also shared by many of these equations has to do with a particular set of solutions known as solitons, which have a distinctly particle-like behavior: a spatially localized profile propagating at a fixed speed when alone, obeying a "nonlinear superposition principle" so many individual solitons can be present in a solution at once, and maintaining their individual identities after interacting. In order to demystify these peculiar solutions, we'll embark on a mathematical journey that will take us to some unexpected places. To survive, you'll need your wits about you, specifically, most of your knowledge from your first-year graduate physics courses. You have been warned!