  The formation of disk galaxies is one of the most outstanding problems in
modern astrophysics and cosmology. We review the progress made by numerical
simulations carried out on large parallel supercomputers. Recent progress stems
from a combination of increased resolution and improved treatment of the
astrophysical processes modeled in the simulations, such as the
phenomenological description of the interstellar medium and of the process of
star formation. High mass and spatial resolution is a necessary condition in
order to obtain large disks comparable with observed spiral galaxies avoiding
spurious dissipation of angular momentum. A realistic model of the star
formation history. gas-to-stars ratio and the morphology of the stellar and
gaseous component is instead controlled by the phenomenological description of
the non-gravitational energy budget in the galaxy. We show that simulations of
gas collapse within cold dark matter halos including a phenomenological
description of supernovae blast-waves allow to obtain stellar disks with nearly
exponential surface density profiles as those observed in real disk galaxies,
counteracting the tendency of gas collapsing in such halos to form cuspy
baryonic profiles. However, the ab-initio formation of a realistic rotationally
supported disk galaxy with a pure exponential disk in a fully cosmological
simulation is still an open problem. We argue that the suppression of bulge
formation is related to the physics of galaxy formation during the merger of
the most massive protogalactic lumps at high redshift, where the reionization
of the Universe likely plays a key role. A sufficiently high resolution during
this early phase of galaxy formation is also crucial to avoid artificial
angular momentum loss (Abridged).
