  The current status of numerical simulations of galaxy formation is reviewed.
After a short description of the main numerical simulation techniques, three
sample applications illustrate how numerical simulations have provided deeper
insight in the galaxy formation process and how they have illuminated success
and failure of the hierarchical galaxy formation paradigm: N-body simulations
demonstrate that the density profiles of dark matter halos that form in
hierarchical clustering scenarios follow a characteristic law. A comparison
with the kinematics of disk galaxies however unravels that these density
profiles are too concentrated. Hydrodynamical simulation show that the highly
irregular velocity field of merging subclumps at redshift $z\approx 3$ can
easily account for the observed asymmetry in the absorption profiles of low
ionization species in damped \Lya absorption systems. The built-up of galaxies
due to mergers is however also cause for one of the major inconsistencies of
hierarchical structure formation models, the failure to reproduce the sizes of
the present day disk galaxies due to excessive transport of angular momentum
from the baryonic to the dark matter component. Hydrodynamical simulations that
include star formation show that scaling laws like the Tully-Fisher relation
can readily be reproduced in hierarchical scenarios, however the high
concentration of dark matter halos results in a zero-point of the simulated
Tully-Fisher relation that is incompatible wi th observations.
