  We discuss the generalized Newton-Cartan geometries that can serve as
gravitational background fields for particles and strings. In order to enable
us to define affine connections that are invariant under all the symmetries of
the structure group, we describe torsionful geometries with independent torsion
tensors. A characteristic feature of the non-Lorentzian geometries we consider
is that some of the torsion tensors are so-called `intrinsic torsion' tensors
that cannot be absorbed in any of the spin connections. Setting some components
of these intrinsic torsion tensors to zero leads to constraints on the
geometry. For both particles and strings, we discuss various such constraints
that can be imposed consistently with the structure group symmetries. In this
way, we reproduce several results in the literature.
