  This paper develops moving frame theory for partial difference equations and
for differential-difference equations with one continuous independent variable.
In each case, the theory is applied to the invariant calculus of variations and
the equivariant formulation of the conservation laws arising from Noether's
theorem. The differential-difference theory is not merely an amalgam of the
differential and difference theories, but has additional features that reflect
the need for the group action to preserve the prolongation structure.
Projectable moving frames are introduced; these cause the invariant derivative
operator to commute with shifts in the discrete variables. Examples include a
Toda-type equation and a method of lines semi-discretization of the nonlinear
Schr\"odinger equation.
