  Maxwell's Electrodynamics admits two distinct Galilean limits called the
Electric and Magnetic limits. We show that the equations of motion in both
these limits are invariant under the Galilean Conformal Algebra in D=4, thereby
exhibiting non-relativistic conformal symmetries. Remarkably, the symmetries
are infinite dimensional and thus Galilean Electrodynamics give us the first
example of an infinitely extended Galilean Conformal Field Theory in D>2. We
examine details of the theory by looking at purely non-relativistic conformal
methods and also use input from the limit of the relativistic theory.
