This thesis covers a set of works pertaining to the generation and the characterization of structured waves defined by exotic topologies. It first presents a method to fabricate devices that can be used to arbitrarily shape the wavefronts of optical waves by means of a geometric phase. These devices can be used to shape the transverse polarization pattern of a light beam as well. Two new extensions to characterization schemes known as orbital angular momentum (OAM) sorters are then introduced and demonstrated. The first extension consists of a sorting scheme able to characterize both the OAM and the polarization content of an optical wave. As demonstrated, this feature could be of use in high-dimensional quantum cryptography. The other extension consists of an OAM sorter for electron waves whose use in materials science is also demonstrated by employing it to characterize a magnetic structure. A proposal on how to measure the OAM carried by an electron by minimally perturbing it is also discussed. The thesis then moves on towards works describing more exotic types of structured waves. On one hand, it explores the stability of space-varying polarized light beams upon propagation through a nonlinear medium. Namely, their propagation is found to be more stable than what is experienced by beams with phase singularities. On the other hand, the effect of twisting a neutron’s wavefunction is also explored and is suggested to affect some of its electromagnetic properties. Finally, a method used to knot the transverse polarization profile of optical beams is presented. The structure of these optical polarization knots is then accurately characterized to reconstruct some of its topological features.