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. These simulations model the formation of disk galaxies within the current structure formation paradigm in which the Universe is dominated by a cold dark matter component and a cosmological constant. We discuss how computer simulations have been an essential tool in advancing the field further over the last decade or so. 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. We argue that 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. This includes the energy injection by supernovae explosions as well as by accreting supermassive black holes at scales below the resolution. We continue by showing 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 and spurious bulge formation. Finally, we discuss the role of mergers in disk formation, adiabatic halo contraction during the assembly of the disk, cold flows, thermal instability and other aspects of galaxy formation, focusing on their relevance to the puzzling origin of bulgeless galaxies.