The origin of the galaxies represents an important focus of current cosmological research, both observational and theoretical. Its resolution involves a comprehensive understanding of star formation and evolution, galaxy dynamics, supermassive black holes, and the cosmology of the very early universe. In this paper, I will review our current understanding of galaxy formation and review some of the challenges that lie ahead. Specific issues that I address include the galaxy luminosity function, feedback by supernovae and by AGN, and downsizing. I argue that current evidence favours two distinct modes of star formation in the early universe, in order to account for the origin of disk and massive spheroidal galaxies. However perhaps the most urgent need is for a robust theory of star formation.