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Great strides have been made in the last two decades in determining how galaxies evolve from their initial dark matter seeds to the complex structures we observe at z=0. The role of mergers has been documented through both observations and simulations, numerous satellites that may represent these initial dark matter seeds have been discovered in the Local Group, high redshift galaxies have been revealed with monstrous star formation rates, and the gaseous cosmic web has been mapped through absorption line experiments. Despite these efforts, the dark matter simulations that include baryons are still unable to accurately reproduce galaxies. One of the major problems is our incomplete understanding of how a galaxy accretes its baryons and subsequently forms stars. Galaxy formation simulations have been unable to accurately represent the required gas physics on cosmological timescales, and observations have only just begun to detect the star formation fuel over a range of redshifts and environments. How galaxies obtain gas and subsequently form stars is a major unsolved, yet tractable problem in contemporary extragalactic astrophysics. In this paper we outline how progress can be made in this area in the next decade., 8 pages, Science white paper for the Astro2010 Decadal Survey, 2 color figures