Ultracold atomic gases in optical lattices: mimicking condensed matter physics and beyond

We review recent developments in the physics of ultracold atomic and
molecular gases in optical lattices. Such systems are nearly perfect
realisations of various kinds of Hubbard models, and as such may very
well serve to mimic condensed matter phenomena. We show how these
systems may be employed as quantum simulators to answer some challenging
open questions of condensed matter, and even high energy physics. After
a short presentation of the models and the methods of treatment of such
systems, we discuss in detail, which challenges of condensed matter
physics can be addressed with (i) disordered ultracold lattice gases,
(ii) frustrated ultracold gases, (iii) spinor lattice gases, (iv)
lattice gases in ``artificial'' magnetic fields, and, last but not
least, (v) quantum information processing in lattice gases. For
completeness, also some recent progress related to the above topics with
trapped cold gases will be discussed.