  In this paper, we discuss Galilean relativistic Proca theory in detail. We
first provide a set of mapping relations, derived systematically, that connect
the covariant and contravariant vectors in the Lorentz relativistic and
Galilean relativistic formulations. Exploiting this map, we construct the two
limits of Galilean relativistic Proca theory from usual Proca theory in the
potential formalism for both contravariant and covariant vectors which are now
distinct entities. An action formalism is thereby derived from which the field
equations are obtained and their internal consistency is shown. Next we
construct Noether currents and show their on-shell conservation. We introduce
analogues of Maxwell's electric and magnetic fields and recast the entire
analysis in terms of these fields. Explicit invariance under Galilean
transformations is shown for both electric/magnetic limits. We then move to
discuss Stuckelberg embedded Proca model in the Galilean framework.
