  An overview over the current status of modeling galaxies by means of
numerical simulations is given. After a short description of how galaxies form
in hierarchically clustering scenarios, success and failures of current
simulations are demonstrated using three different applications: the morphology
of present day galaxies; the appearance of high redshift galaxies; and the
nature of the Ly-alpha forest and metal absorption lines. It is shown that
current simulations can qualitatively account for many observed features of
galaxies. However, the objects which form in these simulations suffer from a
strong overcooling problem. Star formation and feedback processes are likely to
be indispensable ingredients for a realistic description even of the most basic
parameters of a galaxy. The progenitors of todays galaxies are expected to be
highly irregular and concentrated, as supported by recent observations. Though
they exhibit a velocity dispersion similar to present day L > L^* galaxies,
they may be much less massive. The filamentary distribution of the gas provides
a natural explanation for Ly-alpha and metal absorption systems. Furthermore,
numerical simulations can be used to avoid misinterpretations of observed data
and are able to alleviate some apparent contradictions in the size estimates of
Ly-alpha absorption systems.
