  The paper is devoted to the study of the essential spectrum of discrete
Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit
operators method. This method has been applied by one of the authors to
describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger
operators, square-root Klein-Gordon operators, and Dirac operators under quite
weak assumptions on the behavior of the magnetic and electric potential at
infinity. The present paper is aimed to illustrate the applicability and
efficiency of the limit operators method to discrete problems as well. We
consider the following classes of the discrete Schr\"{o}dinger operators: 1)
operators with slowly oscillating at infinity potentials, 2) operators with
periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are
discrete quantum analogs of the acoustic propagators for waveguides; 4)
operators with potentials having an infinite set of discontinuities; and 5)
three-particle Schr\"{o}dinger operators which describe the motion of two
particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.
