  We examine three versions of non-relativistic electrodynamics, known as the
electric and magnetic limit theories of Maxwell's equations and Galilean
electrodynamics (GED) which is the off-shell non-relativistic limit of Maxwell
plus a free scalar field. For each of these three cases we study the couplings
to non-relativistic dynamical charged matter (point particles and charged
complex scalars). The GED theory contains besides the electric and magnetic
potentials a so-called mass potential making the mass parameter a local
function. The electric and magnetic limit theories can be coupled to twistless
torsional Newton-Cartan geometry while GED can be coupled to an arbitrary
torsional Newton-Cartan background. The global symmetries of the electric and
magnetic limit theories on flat space consist in any dimension of the infinite
dimensional Galilean conformal algebra and a $U(1)$ current algebra. For the
on-shell GED theory this symmetry is reduced but still infinite dimensional,
while off-shell only the Galilei algebra plus two dilatations remain. Hence one
can scale time and space independently, allowing Lifshitz scale symmetries for
any value of the critical exponent $z$.
