  We study the asymptotic behavior of solutions of discrete nonlinear
Schr\"odinger-type (DNLS) equations. For a conservative system, we consider the
global in time solvability and the question of existence of standing wave
solutions. Similarities and differences with the continuous counterpart
(NLS-partial differential equation) are pointed out. For a dissipative system
we prove existence of a global attractor and its stability under finite
dimensional approximations. Similar questions are treated in a weighted phase
space. Finally, we propose possible extensions for various types of DNLS
equations.
