  A detailed and systematic formulation of Carrollian relativity is provided.
Based on the transformations, first provided by Sengupta [19], we construct a
mapping between Lorentz relativistic and Carrollian relativistic vectors. Using
this map the Carroll theory is built from the standard Maxwell action. We show
that we get self-consistent equations of motion from the action, both in
electric and magnetic limits. We introduce Carroll electric and magnetic
fields. A new set of maps is derived that connects Carroll electric and
magnetic fields with the usual Maxwell ones and yields Carroll equations in
terms of fields. Consistency of results with the potential formulation is
shown. Carroll version of symmetries like duality, gauge, shift, Noether and
boost are treated in details and their implications elaborated. Especially,
boost symmetry provides a link to the various maps used in this paper.
