  We investigate the symmetry structure of the non-relativistic limit of
Yang-Mills theories. Generalising previous results in the Galilean limit of
electrodynamics, we discover that for Yang-Mills theories there are a variety
of limits inside the Galilean regime. We first explicitly work with the $SU(2)$
theory and then generalise to $SU(N)$ for all $N$, systematising our notation
and analysis. We discover that the whole family of limits lead to different
sectors of Galilean Yang-Mills theories and the equations of motion in each
sector exhibit hitherto undiscovered infinite dimensional symmetries, viz.
infinite Galilean Conformal symmetries in $D=4$. These provide the first
examples of interacting Galilean Conformal Field Theories (GCFTs) in $D>2$.
