  We revisit the formulation of non-relativistic (NR) string theory and its
target space geometry. We obtain a new formulation in which the geometry
contains a two-form field that couples to the tension current and that
transforms under string Galilei boosts. This parallels the Newton-Cartan
one-form that couples to the mass current of a non-relativistic point particle.
We show how this formulation of the NR string arises both from an infinite
speed of light limit and a null reduction of the relativistic closed bosonic
string. In both cases, the two-form originates from a combination of metric
quantities and the Kalb-Ramond field. The target space geometry of the NR
string is seen to arise from the gauging of a new algebra that is obtained by
an Inonu-Wigner contraction of the Poincar\'e algebra extended by the
symmetries of the Kalb-Ramond field. In this new formulation, there are no
superfluous target space fields that can be removed by fixing a St\"uckelberg
symmetry. Classically, there are no foliation/torsion constraints imposed on
the target space geometry.
