  We find a Polyakov-type action for strings moving in a torsional
Newton-Cartan geometry. This is obtained by starting with the relativistic
Polyakov action and fixing the momentum of the string along a non-compact null
isometry. For a flat target space, we show that the world-sheet theory becomes
the Gomis-Ooguri action. From a target space perspective these strings are
non-relativistic but their world-sheet theories are still relativistic. We show
that one can take a scaling limit in which also the world-sheet theory becomes
non-relativistic with an infinite-dimensional symmetry algebra given by the
Galilean conformal algebra. This scaling limit can be taken in the context of
the AdS/CFT correspondence and we show that it is realized by the `Spin Matrix
Theory' limits of strings on AdS$_5$ $\times$ $S^5$. Spin Matrix theory arises
as non-relativistic limits of the AdS/CFT correspondence close to BPS bounds.
The duality between non-relativistic strings and Spin Matrix theory provides a
holographic duality of its own and points towards a framework for more
tractable holographic dualities whereby non-relativistic strings are dual to
near BPS limits of the dual field theory.
