  An overview is given of basic models combining discreteness in their linear
parts (i.e. the models are built as dynamical lattices) and nonlinearity acting
at sites of the lattices or between the sites. The considered systems include
the Toda and Frenkel-Kontorova lattices (including their dissipative versions),
as well as equations of the discrete nonlinear Schroedinger (DNLS) and
Ablowitz-Ladik (AL) types, and DNLS-AL combination in the form of the Salerno
model. The interplay of discreteness and nonlinearity gives rise to a variety
of states, most important ones being self-trapped discrete solitons. Basic
results for one- and two-dimensional (1D and 2D) discrete solitons are
collected in the review, including 2D solitons with embedded vorticity, and
some results concerning mobility of discrete solitons. Main experimental
findings are overviewed too. Models of the semi-discrete type, and basic
results for solitons supported by them, are also considered, in a brief form.
Perspectives for the development of topics covered the review are discussed
throughout the text.
