  This work summarizes the most important developments in the construction and
application of the Dirac-Moshinsky oscillator (DMO) with which the author has
come in contact. The literature on the subject is voluminous, mostly because of
the avenues that exact solvability opens towards our understanding of
relativistic quantum mechanics. Here we make an effort to present the subject
in chronological order and also in increasing degree of complexity of its
parts. We start our discussion with the seminal paper by Moshinsky and
Szczepaniak and the immediate implications stemming from it. Then we analyze
the extensions of this model to many particles. The one-particle DMO is
revisited in the light of the Jaynes-Cummings model in quantum optics and
exactly solvable extensions are presented. Applications and implementations in
hexagonal lattices are given, with a particular emphasis in the emulation of
graphene in electromagnetic billiards.
