  This paper mainly contributes to the extension of Noether's theorem to
differential-difference equations. For that purpose, we first investigate the
prolongation formula for continuous symmetries, which makes a characteristic
representation possible. The relations of symmetries, conservation laws and the
Fr\'echet derivative are also investigated. For non-variational equations,
since Noether's theorem is now available, the self-adjointness method is
adapted to the computation of conservation laws for differential-difference
equations. A couple of differential-difference equations are investigated as
illustrative examples, including the Toda lattice and semi-discretisations of
the Korteweg-de Vries (KdV) equation. In particular, the Volterra equation is
taken as a running example.
