  We construct explicit examples of non-relativistic supersymmetric field
theories on curved Newton-Cartan three-manifolds. These results are obtained by
performing a null reduction of four-dimensional supersymmetric field theories
on Lorentzian manifolds and the Killing spinor equations that their
supersymmetry parameters obey. This gives rise to a set of algebraic and
differential Killing spinor equations that are obeyed by the supersymmetry
parameters of the resulting three-dimensional non-relativistic field theories.
We derive necessary and sufficient conditions that determine whether a
Newton-Cartan background admits non-trivial solutions of these Killing spinor
equations. Two classes of examples of Newton-Cartan backgrounds that obey these
conditions are discussed. The first class is characterised by an integrable
foliation, corresponding to so-called twistless torsional geometries, and
includes manifolds whose spatial slices are isomorphic to the Poincar\'e disc.
The second class of examples has a non-integrable foliation structure and
corresponds to contact manifolds.
