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We refine and develop the inverse scattering theory on a lattice in such away that the Ablowitz-Ladik lattice and derivative NLS lattices as well astheir matrix analogs can be solved in a unified way. The inverse scatteringmethod for the (matrix analog of the) Ablowitz-Ladik lattice is simplified tothe same level as that for the continuous NLS system. Using the lineareigenfunctions of the Lax pair for the Ablowitz-Ladik lattice, we can constructsolutions of the derivative NLS lattices such as the discrete Gerdjikov-Ivanov(also known as Ablowitz-Ramani-Segur) system and the discrete Kaup-Newellsystem. Thus, explicit solutions such as the multisoliton solutions for thesesystems can be obtained by solving linear summation equations of theGel'fand-Levitan-Marchenko type. The derivation of the discrete Kaup-Newellsystem from the Ablowitz-Ladik lattice is based on a new method that allows usto generate new integrable systems from known systems in a systematic manner.In an appendix, we describe the reduction of the matrix Ablowitz-Ladik latticeto a vector analog of the modified Volterra lattice from the point of view ofthe inverse scattering method.

A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices