  The study of the properties of quantum particles in a periodic potential
subject to a magnetic field is an active area of research both in physics and
mathematics; it has been and it is still deeply investigated. In this review we
discuss how to implement and describe tunable Abelian magnetic fields in a
system of ultracold atoms in optical lattices. After discussing two of the main
experimental schemes for the physical realization of synthetic gauge potentials
in ultracold set-ups, we study cubic lattice tight-binding models with
commensurate flux. We finally examine applications of gauge potentials in
one-dimensional rings.
