  In this paper we review two concepts directly related to the Lax
representations: Darboux transformations and Recursion operators for integrable
systems. We then present an extensive list of integrable
differential-difference equations together with their Hamiltonian structures,
recursion operators, nontrivial generalised symmetries and Darboux-Lax
representations. The new results include multi-Hamiltonian structures and
recursion operators for integrable Volterra type equations, integrable
discretization of derivative nonlinear Schr\"odinger equations such as the
Kaup-Newell lattice, the Chen-Lee-Liu lattice and the Ablowitz-Ramani-Segur
(Gerdjikov-Ivanov) lattice. We also compute the weakly nonlocal inverse
recursion operators.
