String theory and non-relativistic gravity

We identify the proper geometry underlying non-relativistic string theory as 'String Newton-Cartan Gravity'. It has the distinguishing feature that one-dimensional foliation in Newton-Cartan geometry is replaced by a two-dimensional foliation on the target space. We discuss some basic properties of the string Newton-Cartan space-time, in particular it's behaviour under T-duality transformations. This leads to interesting non-relativistic conformal field theories at the boundary that have applications in condensed matter physics.