  Cartan's method of moving frames is briefly recalled in the context of
immersed curves in the homogeneous space of a Lie group $G$. The contact
geometry of curves in low dimensional equi-affine geometry is then made
explicit. This delivers the complete set of invariant data which solves the
$G$-equivalence problem via a straightforward procedure, and which is, in some
sense a supplement to the equivariant method of Fels and Olver. Next, the
contact geometry of curves in general Riemannian manifolds $(M,g)$ is
described. For the special case in which the isometries of $(M,g)$ act
transitively, it is shown that the contact geometry provides an explicit
algorithmic construction of the differential invariants for curves in $M$. The
inputs required for the construction consist only of the metric $g$ and a
parametrisation of structure group SO(n); the group action is not required and
no integration is involved. To illustrate the algorithm we explicitly construct
complete sets of differential invariants for curves in the Poincare half-space
$H^3$ and in a family of constant curvature 3-metrics. It is conjectured that
similar results are possible in other Cartan geometries.
