The aim of this thesis is to introduce my work which has generally been focused on optical wavefronts that have the unusual property of resisting commonplace phenomena such as diffraction and dispersion. Interestingly, these special beams are found both in linear and nonlinear situations. For example, in the linear regime, localized spatio-temporal waves which resemble the spherical harmonic symmetries of the hydrogen quantum orbitals can simultaneously negotiate both diffractive and dispersive effects. In the nonlinear regime, dressed optical filaments can be arranged to propagate multi-photon produced plasma channels orders of magnitude longer than expected. The first portion of this dissertation will begin by surveying the history of diffraction-free beams and introducing some of their mathematical treatments. Interjected throughout this discussion will be several relevant concepts which I explored during my first years a CREOL. The discussion will then be steered into a detailed account of diffraction/dispersion free wavefronts which display hydrogen-like symmetries. The second segment of the document will cover the highly nonlinear process of optical filamentation. This chapter will almost entirely investigate the idea of the dressed filament, an entity which allows for substantial prolongation of this light string. I will then conclude by delving into the topic of supercontinuum generation in parabolic multimode fibers which, in the upcoming years, has great potential of becoming important in optics.