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Numerical simulations of large-scale structure have met with great success. However these same simulations fail to account for several of the observed properties of galaxies. On large scales, ~ 0.01-100 Mpc, the ansatz of cold, weakly interacting dark matter has led to realistic maps of the galaxy distribution, under the assumptions that light traces mass and that the initial conditions are provided by the observed temperature fluctuations in the cosmic microwave background. On smaller scales, light no longer traces mass because of the complexity of galaxy and star formation. Baryon physics must be added to the simulations in order to produce realistic galaxies. It is here that the modelling is still inadequate.

In this review, we will begin with the standard phenomenology of galaxy formation, then discuss methods and present the recent observational and modeling advances, finishing with a summary of the numerous outstanding issues in galaxy formation theory.