  Orbital Angular Momentum (OAM) waves were first recognized as those specific
vortex solutions of the paraxial Helmholtz equation for which the orbital
contribution to the total angular momentum of the beam yields an integer
multiple of $\hbar$ along the propagation direction. However, this class of
solutions can be generalized to include more sophisticated vector vortex waves
with coupled polarization and spatial complexity, that are eigenfunctions of
the third component of the angular momentum operator. In this work, a rigorous
framework is proposed for the analysis of all the possible families of vortex
solutions to the homogeneous Helmholtz equation. Both the scalar and vector
cases are studied in depth, making use of an operator approach which emphasizes
their intimate connection with the two-dimensional rotation group. Furthermore,
a special focus is given to the characterization of the propagation properties
of the most popular families of paraxial OAM beams.
